The problem with the words "capitalism" and "socialism" is that people use them pretty much any way they want, so they're not really too useful. But to give you some perspective, I'm going to show some examples that are at least consistent.
First, capitalism and socialism as economic theories. I'm not going to start with who first said some theory is capitalistic or socialistic - instead, I'm going to jump straight into the great fight between the two, which has been going on about since the start of European industrialization to about the second world war.
Both capitalism and socialism (as economic theories) were meant to explain how to best allocate resources. The problem is: we have some limited resources, and we want to use them in the way that produces as much wealth as possible. There's no "social" overlap here - the only concern is using limited resources to satisfy as many wants as possible, and satisfying more urgent wants before less urgent wants.
The capitalists say: the markets, with their prices continually changing to account for everything that happens in pretty much real time, are an efficient method of valuing the inherently subjective values that people have, as well as all the alternate production methods. That is, just looking at the prices of commodities and capital, you can see whether it makes more sense to (say) invest the capital you have in refining iron sulfide ores, or iron oxide ores. The same way, you can also decide whether it makes more sense to invest in iron production, or (say) wheat production. The more money you can make in any given venture (that is, the difference between the "ingredient" prices and "product" prices), the more urgent the need is, and the more efficient the use of resources. Note that we don't care about the absolute prices - only about the ratio between the cost of inputs, capital, outputs and throughput. That is, the most profitable venture can just as easily be producing a kilo of bread for 1 USD, as producing a single yacht for millions (which of course explains why people bother producing other things than the most expensive ones).
The socialists say: the markets are horribly inefficient. People are competing with each other, wasting lots of wealth in products that weren't sold for their expected value, advertising etc. A central authority could allocate all resources and capital according to the best needs of the people, and avoid all the losses incurred over the competition and uncertainty of the market. As a bonus, since the state would control all the means of production, there would be no profit margins - so instead of shifting the profits to the entrepreneur class, we can distribute them to everyone equally. Finally, since bigger ventures are inherently more efficient, managing production under one central authority will give us unprecedented production efficiency boosts - we can produce ten times as much with the same effort and investment.
The same way, income inequality wasn't about being fair ("should all people have the same income?"); it was about which way actually makes people better off.
The capitalists say: under free market conditions, profits go to the people who best satisfy the most urgent needs of the people. The income inequality is what motivates people to best serve other people - they expend their labor and capital for their own motives, with the result being investment flowing into the needs people find most urgent.
The socialists say: the profit the entrepreneur receives increases the costs of the product. If we remove the entrepreneur and replace him with a central authority (which would either take no profit, or some fixed amount specified by law), the people will be richer. With the productivity increases, we can afford a life where all wants will be satisfied and people will only work at their leisure.
At this point, the approach of both to the idea of happiness is the same - richer people are happier, so we want to make everyone as rich as possible given the resources we have available. The people will then use their part of the wealth in the way they desire, making them as happy as possible.
Both theories are addressing the same problem, and both have their predictions that can be verified in practice, as well as explored in thought. And that's what the main fight between capitalism and socialism was about - which one produces more satisfaction for people. As time went on, both were thoroughly tested. The main argument against socialism was that with no markets, there are no prices which represent the diverse subjective values of every single user of the market and the resources and capital available. The main argument against capitalism was that the markets delegate wealth to people who aren't directly involved in production (e.g. financiers, marketeers, traders, entrepreneurs, capitalists...), and it tends to partition the economy (e.g. you have 20 companies producing steel, each with its own factories, logistics, contracts... while under socialism, there would be just one company that could build the biggest possible factories, with just one company providing transportation etc.).
I'm not going to go into detail here - you can find entire books on the arguments for both the capitalist side and the socialist side. Of course, when socialist theories were tried in practice, it became pretty obvious that the capitalists were right at least in one thing - without the coordination that prices provide, allocation of resources becomes pretty much impossible. Socialists producers kept producing too much of some things, and too little of others; and fixed prices weren't able to cope with changing economic conditions - in capitalism, when some product becomes scarce (e.g. there's a drought that means there's less wheat), prices rise in response. This both encourages further investment (more farmers, buying grain from abroad etc.), and saving (people consume less of the thing that's scarce). In socialism, the same thing would require an edict that sets a new price, and a new plan that reallocates all the resources of the country (remember, all means of production belong to the state) as needed. This is extremely complicated, and was outright out of the question given the state of communications, technology etc. at the time. And keep in mind that socialist societies still had the capitalist countries to copy - the routinely adjusted their prices and production based on what they could find abroad.
So socialism as an economic theory was in a bit of a pickle. Their claimed increases in efficiency turned out to be anything but; indeed, the efficiency dropped so low that people in industrialized countries dropped into starvation, with the worst cases resulting in millions of deaths by starvation. That was something pretty hard to ignore for anyone but the hardest socialists - if socialist methods of production are so much more efficient than capitalist, how come starvation is a rare occurrence in capitalist countries but more common in socialist countries than even in places like czarist Russia? The promised increase in living conditions was based pretty much entirely on the massively increased productivity under socialism - since the reverse turned out to be the case, they needed some other method to promise wealth to the masses.
Of course, they didn't have to think of anything new. There's been hundreds of schools of socialism that had little in common with each other except for the central idea of countries (and sometimes the world) following one central plan. As it was said, "the biggest enemy of a socialist is a socialist with a different plan". But what emerged as a very successful political approach was the merging of capitalism and socialism - the idea that you can rely on capitalism and free markets to maintain most of the economic activity of a society, while imposing laws that regulate and control the markets for the good of the people. Of course, there's also been lots of offshoots that denied various theses of the capitalist schools; some fell back into the old mercantilist/nationalist ways ("free markets are fine, but foreign trade should be restricted"), guilds ("all trades should be regulated"), and many others - I'm going to skip over most of these.
The main idea is that you could use the superior production efficiency of capitalism (which they grudgingly had to admit) with the claimed increases of quality of life under socialism's expected productivity increases. The approaches are mostly interventionist - for example, imposing import fees to encourage domestic production instead of imports, having different taxes for different products and people etc. Each of these laws then has both immediate and secondary(+) effects.
For example, if your domestic steel producers have trouble competing with producers in other countries, the state can fix the import prices of steel at a level competitive to (or outright higher than) domestic prices. The immediate effect is that domestic steel producers will be better off - they'll have higher margins, they can afford to employ more people, invest into more marginal productions etc.; the state gains bonus income to help their spending. But of course, this comes at a cost to everyone else - the prices of steel are higher than they would have been, steel is produced with materials and in areas that aren't best suited to steel production, capital that has already been invested abroad is not going to pay off quite as well etc. And this needs some justification - you need to claim that the cost to everyone using steel (and state services paid for by the tax) is worth the increase to the conditions of the steel producers.
Now, various socialist schools give many different justifications (e.g. the classical nationalistic thought "domestic production is more stable/safer - we can't afford to rely on steel imports!"). But by far the most popular is this one: if we let people use their wealth to the best of their ability and according to their subjective values, they are going to waste it on unimportant things, rather than what's good for the society. So, for example, if we don't provide for people's healthcare, they are not going to provide for it themselves, because while they're still healthy, they will rather spend their money on (say) new shoes. If we don't forbid gambling, people are going to waste all their money in slot machines, while they drop deeper in debt and (say) starve. If we don't set by law how many bakeries there are to be per 1000 people, nobody will build any bakeries.
Suddenly, the socialists seem like the only party that cares about the people at all. Their explicit goals are to make people better, regardless of what the individuals themselves want. Things that are aligned with whatever the lawmaker thinks is better are supported (public education, theaters, healthcare, diesel engines, free bread...); while things that he thinks are bad are forbidden or taxed (alcohol, tobacco, drugs, incandescent light bulbs...).
Where does capitalism stand on this? It doesn't! Capitalism is an economic theory - a way of using the limited resources we have in the best way known possible. It points out the inefficiencies of all these socialist methods (e.g. "quality and/or availability of healthcare will suffer" or "import taxes only help the people they protect, and hurt everyone else - domestic and foreign"). But it doesn't tell you what to do with the wealth once you produce it - that's out of the scope of the problem. Ultimately, every good or bad thing entails a value judgement, and values are inherently subjective. Worse, they change all the time in response to the environment (e.g. a healthy person might be unwilling to pay $100 a month for health insurance, while as soon as he becomes gravely ill, he would be grudgingly willing to pay far more). Capitalism just tries to give everyone as much wealth as possible, in the only way that works on a global scale - satisfying the most urgent needs first. It doesn't try to control people - it just gives them more options. It doesn't try to force some sort of security on people - it gives them the resources to secure themselves, if they find the value trade-off favorable. It doesn't take away your responsibilities; neither does it steal your options.
So finally on the end of the spectrum, all the way to practical politics - socialism decides what's good for you. Capitalism lets you do the decision, for better and worse. If your "leading socialist" decides they want to encourage happiness, and that definition of happiness happens to be the same as yours, you might say that socialism encourages happiness. If it doesn't, you might say that socialism discourages happiness. Ultimately, all it does is take the choice away from you - it makes you follow the central plan, whatever it is.
As a final addendum, it seems that many people think that capitalism encourages competition, while socialism encourages cooperation. That's not true at all - capitalism is built from the ground up on cooperation across all of society. The competition is only a surface feature between people who are, well, competing for their place on the market. But those people are at the same time cooperating with everyone else in the society. Capitalism is not a zero-sum game - cooperation increases value for both parties, it doesn't take from one and give to another. For a trade to occur, you are giving away something of low value for something of high value - and the other party has it the same way. Both of you profit from the trade, because again, values are inherently subjective. That's why trade happens, and that's why trade (and cooperation) increases the wealth for everyone involved.
In contrast, in socialism (of the "not economical theory" kind; in proper terms, this would include movements like interventionism, protectionism, nationalism etc. etc.), everyone is competing for the special favors of the state. Getting the right law signed can make or break a business. If you can make the state award you a monopoly on some product or service (e.g. taxi services in a city), you benefit to the detriment of everyone else. If you ensure that your company gets subsidies, you benefit to the detriment of everyone else. If you ensure that your products receive an import tax, you benefit to the detriment of everyone else. In socialism (and protectionism of any kind), the default state is that everyone is competing with everyone else - and "winning" the competition almost always means getting special privileges for yourself, at cost to everyone else. Various schools of socialism differ in the exact approach taken, but the core remains - to be successful, you need political power. Even if noone in the whole country wants to buy your products, with enough political power you can force them to - the only power the consumers have is in their voting.
The best that can be said for happiness under socialism is that the state can force you to do something. Either you would have done that anyway, because you value that (e.g. you're the kind who would go to a theater at market prices, rather than requiring subsidies) - in which case capitalism would serve that value just as easily (though not necessarily for the same cost) - or you wouldn't. And if you wouldn't, you might find yourself happier - it's definitely possible that going just by your own instincts and value judgments, you'd miss an opportunity to be happy. If you force people to only work two hour days, they might end up happier. Or they might not. They might also drop all the way to joblessness (if their marginal productivity isn't enough to support their lifestyle with just two hours of work a day), which again might be both something that makes you more or less happy.
Ultimately, happiness is just another subjective value. Capitalism has a great track record of satisfying subjective values; socialism has a great track record of suppressing subjective values and replacing them with some design from above. And to see what actually makes you happy, you really need to be able to choose different lives - so you have something tangible to compare. In some ways, socialist societies made that easier (e.g. if engineer's wages are the same in every city, you can more freely move between cities to find which one suites you best); in others, worse (e.g. without the different wages and prices, there's no guidance on where to invest capital etc., so there might e.g. be no flat for you to move in).
Keep in mind that throughout, I was always talking about wealth, not money. Lots of people conflate capitalism with making money (and worse, money with wealth), but that's a serious misunderstanding. Someone living on their own farm, happy and having fun all day, with provisions for old age and illness etc., might very well be wealthier than the person with the most money in the world. Values are subjective, you see. Money is about trust - as it is said, you cannot eat money. If people stop accepting your money, it's utterly worthless. If you shipwreck on a deserted island with a hundred people, free market will tell you how to best use the resources you have to survive. Who brought how much money would have very little impact on the society you form. You will quickly find that only cooperation and division of labor helps you improve your living conditions - and cooperation and division of labor is what free markets are all about. Whether that leads to happiness is up to you, and you get the choice to steer life your way, as long as you're willing to invest in it.