Not sure this is entirely balanced, but here are some pros and cons. Many of my cases come from France, where I used to live and which has long had a Colbert-inspired love affair with state ownership.
This answer is strictly concerned with companies operation in for profit sectors as the OP asked for economic factors such as income and financing. Companies that are run for social purposes or operating in sectors where the market is not necessarily ideal, like health care or education aren't covered. Other answers do that. It also doesn't really concern itself where cases where the state provided initial assistance or subsidies but without ownership and over a short period (arguably, SpaceX).
Innovation (or the lack thereof)
When I lived in France, we had a SOE, Honeywell Bull, who manufactured mainframes. They were the suckiest mainframes, far suckier than the not-great IBM mainframes. But being a state company they were a big fish.
An underperforming private company usually ends up bankrupt. An SOE can remain a zombie.
Sources of revenue (hah!)
Again taking Bull, far from being an extra source of revenue, they gobbled up taxpayer bailouts every so often. Tellingly, while the EU never did put in rules limiting how much governments could finance themselves from their SOE, they did put in fairly strict limits on much they could subsidize their SOEs. Rules that the more SOE-happy countries were always trying to get around.
Or the famous Credit Lyonnais case of the 90s in France.
Crédit Lyonnais is a historic French bank. In the early 1990s it was the largest French bank, majority state-owned at that point. Crédit Lyonnais was the subject of poor management during that period which almost led to its bankruptcy in 1993. It was acquired by former rival Crédit Agricole in 2003.
(look at r13's comment, and do the math)
Face it, no one likes to see mass layoffs, even if it is only to trim an underperforming division and hire in another. When a private company does it, depending on the country, they may get some governmental pushback, but it is a decision taken by an independent actor. When a state-owned company does it, there can be a political cost to its owner, the government. Which might decide not to layoff.
When monopolists, can be expensive and bad for customers
Before some measure of competition got injected, France Telecom was often criticized for delivering iffy phone services at high prices. At one point, their "certified" cassette answering machines were 4x US prices.
British Columbia, where I now live, has a Crown company that has a monopoly that provides car insurance for consumers. Guess what, it also has much higher prices than other provinces. Does it make lots of money for the government? It does not, it is that inefficient.
(see also Fizz's answer for an excellent point, no need for me to repeat it)
Special rules for special companies.
I'll take an example which isn't really an SOE, SNC-Lavalin in Canada. They got caught bribing abroad. Then they almost got a sweetheart deal because they were well-connected with the Canadian government. Big scandal on Trudeau's goverment. Those risks go up, way up, when the state owns the company, but they are tend to be more pervasive the more the state is involved in economic production.
(I'll to find a directly state-owned case of malfeasance forgiven, but it will be obvious to most that the state is unlikely to treat itself unkindly in most cases).
Governments are unlikely to "rock the boat".
Let's take Tesla. When it first sold the Roadster it took a very bizarre spin on EVs: making them sexy. Before Tesla, this just wasn't how EV cars were designed and marketed. EVs were meant to be parsimonious cars for ascetic, high-minded, environmentally-conscious folk: EVs also didn't really exist.
Once a government has stakes in many car manufacturers it will naturally try to limit their competition. This isn't necessarily a monopoly per se, but it will limit the funding and ease of bringing new business models to fruition when the government both regulates and benefits from economic activity in a given economic sector.
Speaking on a related subject, one could consider the taxpayer-funded bits of NASA's contractors on the SLS to be almost SOEs. 20B$ later that turkey hasn't flown and will cost $2B/launch. Probably a comparable sum of cashflow and investment later SpaceX almost has Starship costing maybe as low as 10M+ per launch (yes, yes, that is Musk-speak so that needs a grain of salt, but it's still likely to be ballpark-ish). SpaceX has taken over a huge chunk of the launch market already with the Falcons.
Sucking up financing.
unlimited budget and financing from the government
Well, yes. This means that better alternatives and innovative solutions may be starved of capital. Is that a "pro"???
Fannie Mae & Freddie Mac (quasi-SOEs) and the 2008 financial crash They're not strictly for-profit, having more of a social dimension, but hey, couldn't resist.
Some actual pros:
Under some conditions a new market may be difficult to enter or finance and a government can start a new industry from scratch. EDF, in France, did very well for decades building nuclear reactors and it's hard to see private sector companies lasting it out in that domain (Westinghouse is dead, for example).
As one commenter pointed out, high speed rail is something that the French SNCF has innovated and delivered (so did Japan Rail). Then again, USA's Amtrak shows that state ownership and rail is not an automatic win.
Some sectors are more naturally suited to nationally or regionally coordinated/unitary infrastructures / monopoly providers - roads, rail, utilities, airports... etc. The state may operate those better than private sector companies that are unsuitably regulated. Strangely enough France has a very good network of high speed highways, built and run by private operators. It also has some world-class private companies specialized in municipal water delivery, something where public ownership isn't always good. There is a whole concept of involving private actors in these types of projects, PPP, that doesn't always deliver when private companies are brought in.
In the US, there has been a push by some municipalities to put together their own municipal broadband services, citing bad service by the private sector. Of course, any protests by the much-beloved US telecoms companies, which has included getting state laws forbidding such initiatives passed, are purely in the interest of users.
Or an endeavor may take a long time to come to fruition, yield massive, but diffuse economic benefits and needs government nurturing. Companies/government departments during agricultural research into better crops are something to consider, ex: Embrapa in Brazil.
In a developing country with weaker financial and technological depth, a government can seize an opportunity and quickstart an industry the country has a special advantage in.
In a situation with national resources sometimes it is better to have a national company than leave it to the foreign multinationals. Saudi Arabia is probably better off with Aramco than by letting Shell and Exxon take over (as a comment states, maybe Statoil in Norway is a even better showcase, but that might give the impression that you need a super well-run host country to pull that off).
On the flip side, you have Venezuela's "management" of PDVSA (take the 25 year graph).
A state may also choose to subsidize its SOEs exports to undercut at a loss local providers in its export markets, something known as dumping. This is something that comes up every so often with regards to China, whether that is actually the case or is just a convenient complaint by those local providers. This might be especially true in the case of "strategic industries" where a private sector actor would not have a benefit in behaving thus.
China has been playing that game with rare earth metals.
Success is not impossible...
(assuming there is not too much corruption)
But in many cases the more likely outcome will be that an SOE will underperform, economically. So even a state that is successfully deploying SOEs for specific purposes will likely need a larger stable of private enterprises to provide the tax base necessary.
Most of China's newfound dynamism, the companies we hear about, aren't from an SOE background. Alibaba TikTok
2022 Paper abstract re China SOE:
The nominal performance of SOEs is not high enough. Indeed, from 2001 to 2009, the average return on equity of SOEs was 8.16%, while that of non-state owned industrial enterprises was 12.9%. In 2009，the return on equity of SOEs was 8.18%, while it was 15.59% for non-stated owned industrial enterprises. Moreover, the recorded performance of SOEs is not a reflection of their real performance, but the result of numerous preferential policies and an unfair business environment.
p.s. if you feel that I have misrepresented poor SOEs and undersold their benefits that's a cue to write your own answer. I have tried to show cases where for-profit SOEs can be useful - that list got progressively longer and detailed - but consider that to be very much the exception.