TL;DR: There is no simple and concise way... Sorry.
Like even when these terms originated in their modern sense about 200 year ago, there were already countless of definitions of these terms which sometimes were synonyms sometimes, different, often overlapping and so on. Also literally "commune" and "social (society)" aren't really all that different so depending on the language either term could have taken the cake...
Essentially the liberal Revolutions like the French Revolution with their claims of "Liberty, Equality, Brotherhood" got a lot of people inspired, but also somewhat disappointed. As the hyper-individualism essentially meant that the "liberty" still only applied to the rich and the rest of society was largely still where they were unequal and not exactly treated like (close) family either...
So lots of people got thinking about how a better society could look like which you could all somewhat gather under socialism. But who could vary on reform, revolution, state or stateless, commonly owned resources or personalized, markets or share economies and so on and on and on.
Though apparently it was plenty obvious that the new ordering system of society was no longer a church mandated social hierarchy of hereditary monarchs, but a capital based system where the owners of the means of production where the new rulers.
As a consequence of that socialism was largely the demand that those means of production should be owned by those who work them. Either each their own means of production or larger ones by worker collectives.
One variation of that would be communism where you'd build a society around that idea, where the community owns the means of production and conversely everyone owns them equally thus everyone shares the political power that arises from that ownership equally and people share the workload and outcome according to their needs and abilities.
Which is obviously utopian in an age where kings and queens are still a thing, where the means of productions are owned by a handful of rich people and where workers don't produce their own outcome but have to prostitute themselves (sell their bodies, workforce or mind) for a rich person or... die.
And which comprises the ideas of political liberation and self-determination up to the point of reclaiming anarchism as a positive (the deliberate lack of leadership in favor of self-determination of the individual and mutual groups of individuals), the idea of bringing democracy and republican ideas to the workplace (which to this day often follow a model that is closer to monarchism, aristocracy or in the worst case tyranny rather than democracy) and which gains lots of traction and was partially synonymous with the general workers movement, because you know early capitalism really sucked for workers. Like look up how long the battle for the 8 hour work day had been, that healthy work environments weren't really a thing, that without health insurance wasn't a thing, retirement wasn't a thing, even building a deposit of money at the end of the month might not have been a thing. So you wage slaved to your grave and it became harder with every day, also due to child labor competing with your job...
So mixture of tl;dr: Quest for political self-determination society in particular for the groups forgotten by classical liberalism (the vast majority...), democratization of the workplace, worker movements in general.
So those are the rough outlines mid 19th century, but again utopian idea and literally hundreds if not thousands of ideas on specifics and how to implement that.
Then a guy named Marx entered the scene and wrote a manifesto for one of those obscure socialist groups somewhat claiming the label "communist". Delivering an analysis of history based not on king a did stuff b, but ignoring specifics and sketching it only on the basis of how the structure of society follows the mode of production. So according to him agriculture and the limited means to control people created feudalism, where people were so far apart from each other that they had autonomy over their workplace but were busy all day everyday with their survival and where ownership of farm land was the source of power. To capitalism where exchange of goods and services was the power that you could use to gain more goods and services and thus power and where the source of that power is to be able to build commodities and where the road to that is having the means of production.
Long story short he thought society would progress with it's production and that it would basically inevitably go from feudalism to capitalism to communism and from rule of few to rule of more to rule of all (or no rule at all). So Marx wasn't really an enemy of capitalism, he just thought of it as one necessary stepping stone, better than feudalism in some regards, worse in others, but in general lots of room for improvement.
And similar to the liberal revolution he thought the next revolution would be by the proletariat, so the industrial workers, who would soon be the majority, who would be able to work those machines and who would be able to create a society of their own with the need to subjugate another class. Unlike the kings who needed peasants and the capitalists who needed workers, they could literally DIY.
Though again, for Marx capitalism was necessary to develop the means and the mode of production, a premature revolution in peasant society might change who is king but would not change the mode of production or the makeup of society as people would still work 24/7 not having time for politics and self-determination.
Which lots of people found pretty cool and called themselves Marxists much to the dismay of Marx himself who apparently spend quite some time roasting other socialist because they didn't get his ideas and didn't share his passion for actually gaining political power for the working class and forming a more equal society not just better working conditions. Again that was a huge issue that likely has taken up more space than the ideology stuff in many programs...
So he was a fierce opponent of those trying to be on friendly terms with those in power, who gave them improved working and living conditions, but kept the distribution of the means of production and the political disenfranchisement. Though which still might have called themselves socialist, because, you know workers movement, "social" or maybe still in for the political power but more of reformers.
So communists according to Marx (again that term predates Marx): Those in who work to establish the communist (which by definition is also socialist), by organizing in a political mass movement of the working class to seize the means of production and thus overcome the class division and usher in a new era of self-governance. He largely left open how that would look like because self-determination.
Now not much happened for the foreseeable future. The French pivoted between republic and monarchy, Europe mostly took a step backwards and restored absolutism and when Paris tried a commune in 1871 during the war with Germany, it was shortlived, chaotic and gunned down. Which apparently prompted Marx to better organize that shit, which the anarchists within his circle were not a fan of, so he bullied them out of that until people thought he was going to much into authoritarian and also quit and "the first international" collapsed.
Though in the meantime and after that workers movements were getting tracking, things often headed towards revolution and even conservatives like Bismark implemented the first social programs (healthcare, retirement, social security, etc) just to defuse the situation and take the wind out of the sails of the socialists.
So again depending on whom you ask that is either a major accomplishment of socialism, fighting the man until they give them what they want or it's a pyrrhic victory as it meant a decrease in revolutionary potential while still being a long way from democratization of the political and economic landscape.
Now again Marx: Mass movement -> do revolution -> get control of the means of production (dictatorship = government) of the proletariat -> transition into communism.
However while Marx apparently used that dictatorship of the proletariat a whopping 4 times in his writing and basically in the sense of government of the workers, there were others that had the idea of "one would just need to seize government power and then transform top down". Marx's fellow Engels was already not a fan and argued that failed in the Paris Commune already.
Nevertheless in early 20th century Russia, Lenin convinced the social democratic party of Russia to exactly such a plan given that Russia didn't even have a successful liberal revolution, was lacking behind in industrialization and that the most industrial countries weren't developing according to Marx. So he formed a vanguard party and called it Bolsheviks (="majority" after the one election he was able to get one), burned through several fake identities and got himself and his professional revolutionary vanguards imprisoned and exiled... Missing out on the 1905 revolution (which established a parliament, but which was stuffed by the czar and thus made insignificant), the 1917 revolution that got rid of the czar and only after that was able to return to Russia with the help of the German military who though bringing trouble to an enemy country would help them with the war (partially successful, Russia is the only WWI ally who lost the war).
So Lenin fired against anybody, presented himself as supporter of the soviets (not a group but an organization form of direct democratic worker councils, which organized the revolution of 1905 and 1917) something like this: https://en.wikipedia.org/wiki/Democratic_confederalism
He dropped out of WWI and held election, but as he didn't like the results he seized power regardless of that and engaged in a several year long civil war. But as Russia wasn't satisfied with the war and the situation anyway he was still able to win that civil war. Though not with a mass movement, but with a minority government, that engaged in state terrorism to secure it's power and which implemented capitalism with some social security and whatnot to attract foreign grands and industrialize further (yeah by a communist party...). Still the U.S. and capitalism was majorly scared of dissatisfied workers rebelling.
Meanwhile in Germany, the end of WWI also faced a socialist revolution. Though this time the social democrats teamed up with the conservatism and the protofacist parts of the army and gunned it down. They nevertheless managed to force them into a republic and not a constitutional monarchy and some more democratic and social advancements, a modern constitution and lots of other stuff that would have been great for the time, ... unfortunately progressively fewer people were down for that as the left felt cheated by their former social democratic allies and the right still thought that this whole "Marxist" nonsense was to be disposed of and so the system was struggling from day one, the communists fought the social democrats, the conservatives tried to bring the system down and ultimately the fascists profited from all of that, the rest is a sad part of history...
Though in the west this model of social democracy and trying to amend the system, fighting for better conditions and participation but not necessarily revolutionary and through redistribution often was already branded by conservative parties in waves of "red scare" as "SOCIALISM!". So in some countries socialism is ... "just not being in favor of social darwinism and weeding out the poor and working class"....
Meanwhile in Russia Lenin died of a stroke and a charming fellow called Stalin took over. He developed Lenin's power grap, the terror and attempt to industrialize into Stalinism and called it Marxism-Leninism. Ended that capitalism and the idea of a world revolution and advanced the idea of socialism in one country, putting all means to industrializing his counters, argued that Lenin was right and that dictatorship of the proletariat really means dictatorship (by him), he also used his knowledge as a former theology student and replaced Jesus pictures with himself and developed a cult of personality and he also redefined terms, so "socialism" is now the transitioning state between capitalism and communism, which is when the communist party has government control and used that to lay the foundation of communism in countries not yet able to transitions because of a lack of development.
That is basically the blueprint the USSR brought to many other countries which are so called "communist". Which is in quotes because if you recall the definition of communism from earlier or even the one you find on wikipedia of a classless stateless society in which people people own the means of production, produce for their own consumption and share the workload and outcome based and contribute according to their abilities and receive according to their needs... Well there's some distinct discrepancy between that and whatever the hell these countries were doing.
Though given the U.S. and capitalists where still massively scared of workers being dissatisfied with their subpar situation under capitalists labor system, they nonetheless never got tired of calling that communism and labeling everything not in favor of "work or die" as "socialism"/"communism".
Though depending on what perspective you take when looking at it. This "communism" either dragged poor and underdeveloped countries out of poverty and helped them industrialize. Seriously many countries were in such a bad shape that there was only one direction in which any government could have been going. Or you could compare them with western countries and argue they are still largely underdeveloped dictatorships, but also the kind of dictatorships that you don't want to trade with as they would like keep your investment for their people rather than suppressing their people and selling the mining rights to western countries... Just kidding they also suppressed their people massively but yeah weren't that keen on selling mining rights to western countries.
So yeah depending on whom you ask "communism" is either that ideal of a mutual community of people living free and equal sharing co owning the means of production and sharing their output or and that is unfortunately also very likely either whatever a raging conservative doesn't like or a soviet union hardcore fan who thinks the appropriate reaction to democratic protests are to send in tanks...
While "socialism" ... is essentially the same + some social democrats, so incremental social progress, better working conditions, pushing for more democratic and social participation of working class and low income people and so on.
So whether there is a distinction or not depends largely on the context and what the "No I don't want [that]... I want [this]" concretely refers to.
Could be it's the same thing, could be something vastly different, again even 200 years ago people could largely agree on the goals and still spend their time roasting each other for how utopian their ideas are...
So TL;DR There is no simple and concise answer, that largely depends on the context.