Depended really on a whole variety of things.
How respected were you as an artist internationally. (So, was the state interested in "assimilating" you into the narrative)
How revolutionary were you. (CAN you be assimilated into the new Stalinist narrative)
How hotheaded were you. (Are you willing to compromise with the state)

The Soviets wanted to show how "forward thinking" they were, so they didn't just outright kill everyone they hated or deemed formalist/bourgeois/Trotskyite/Menshevik/Fascist/Reactionary.
If the artist was useful for the system and was willing to compromise, then they'd ask for a few propaganda pieces, and would let him write or compose whatever else they wanted on a smaller scale.

The doctrine of Marxism-Leninism boasted that the USSR's culture was the next step in cultural evolution. It was beyond the west. So obviously, assimilating contemporary art was important. Turning avant garde artists like Shostakovich and Mayakovsky into cogs in the system. Shostakovich gave up, Mayakovsky choose to have a bit more dignity/wasn't smart enough/willing to try undermining the system from the inside, so he bit the bulett. Admirable I'd say. That is, if we can believe the reports that it was a suicide.
There is this old joke about the event.

>Is it true that Mayakovsky the poet committed suicide?
>Yes. We even know his last words!
>So?
>"Comrades, don't shoot!"

Basically, if you weren't willing to cooperate, they'd at first stop giving you commissions, and would stop publishing you. Then they'd ban your already published works. (You could still try getting new stuff published and approved, it's just that you old works are now out of print until a new Vozhd comes around and softens the system.
Next is being publicly reprimanded by the Association of Proletarian Writers or the Association of Proletarian Musicians.
Then you get sentenced to 10 years and you're off to Siberia.

>For how long did you get sentenced?
>Ten.
>What for?
>Nothing!
>Come on, don't lie! For nothing you get at least 20!

And even if your sentence is over, they'd just randomly extend it, just to keep you away from the other intellectuals.
If you are proving to be a big nuisance for the Party and the State, then comes the NKVD and takes you in the middle of the night, and shoots you like a dog.

The soviets basically only killed you if you were not up to date with the current party line, and you were unwilling to compromise. (But if you compromised, then the state would compromise too. Shostakovich had to write extravagant, neoromantic/neoclassical symphonies for the Party, but they let him compose chamber works rather freely.)

The Nazis on the other hand wanted to steer the ship in a completely different direction, trying to rectify things of the past. For this reason, the Nazies focused more on assimilating the thinkers of yesterday, like Nietzsche and Wagner.
Why bother with the avant garde at all? It's Jewish rubbish anyway! Romanticism and classicism is the Aryan man's art direction! Sacrifice! The Will overcoming physical bounds! To bring by the end of history purely through will! Pick up that sword, oh, Siegfried and crush this decadent world order!
If you look at it, it's in line with how the regime of Napoleon worked for example. It also supported classicism.
The Nazis wanted to push the reset button on modernity, the Soviets wanted to show they are thinking ahead of capitalism.

These two books paint a good picture of the artistic life in the USSR. They usually intertwine with then-contemporary literature, so it's not only about music.

You can find similarities and analogies between every two objects in classifying them as widely as you can. You can divide them onto one broader category and be happy with it, arguing its validity with analogies popped up. It's all about intellectual jerking off, at some point.

The soviet saw the censorship coming right out of the top establishment (it wasn't a socialistic state in terms of power-sharing, to begin with) by way of denoting anything with a story different from depicting regular agricultural people talk about the revolution or key figures in the mythology and ideology of the new state.
Stalin in 1929, afaik, ordered to outlaw everything that was "formalism" which formally meant everything that highlighted a "form" more than content which meant everything apart from ideologically driven excitement about new life or events directly preceded and influenced it. Since about then, he had decided to be a great sole movie producer taking decisions on what films were to be shown. Of course, it narrowed down the soviet cinema output to 4-5 movies per year, and this span of time is widely regarded as a period of, at least, darkness. Even eager adherents are not prone to name a couple of films of oh-yes-superior soviet culture off that age. Similar courses of action were applied to different strands of art save for the differing executors.
So it started off as a way of repressing the culture which, in this regard, was aimed at gaining insight from the revolution and could bring some undesired questions into peoples' heads.
The establishment had seen them as a bunch of variables one might toggle with. And tools like that were applied mainly to ensure the power of the stalin's government. Films he chose, for instance, were largely patristical depicting him as THE father.
The artistic freedom of ussr in the 20s witnessed the greatest innovative use of montage in cinema, musical lift and, by inertia, a consequent tradition of russian literature. People felt they were not strapped anymore by the framework of proper art and were highly experimental yielding some fruits. The only thing is, theme was politized after the lenin's goverment "nationilized" everything to subsidize every field of the economics.

As for the nazis, they and the soviets in fact pursued the same ideals of having an equal layer of society with a higher order to control them. So, provided they were from the same epoch against almost the same background of ideas, they employ nearly the same means to make up programs to achieve the ideals. But when it boils down to the realisation, the main features were not excessively the same. Although in the nation problem, you can draw parallels as the newly formed USSR set out to erase the "great russian shoevinism" as they put it, nazis wished to force people think about the greatness of their heritage, the great germanic identity in simplistic, similar to the soviets forms of genre art rather than how they have overcome the tsar and lived a wonderful life. So, there's a subtle yet significant difference. The nazis wished to amplify feelings of being born german, the soviets wished to amplify feeling of being born to witness the falldown of the tsardom.
Roughly non-detailed speaking, it was like that.

Soviet and Nazi censorship were very similar in the sense that the state authorities were entitled to ban everything (music, literature,
paintings, cinema, massmedia etc.) that they considered opposing to the state and the state's official ideology.

Specifical differences were caused by the different ideologies, the different historical developments (Nazis 12 years/ commies 70 years) and the different structure of the administrative machineries in both states.

For instance in Nazi-Germany the censorship was executed by the Ministry of Public Enlightenment and Propaganda while the USSR had a government agency engaged with the censorship of a certain field, for instance one agency that censored all publications that were printed.
The film industry was supervized by Stalin himself, if he thought a certain film's message would collide with the goals of the socialist
state or undermine its authority the film was banned, sometimes also the director purged from the party or arressted and sent to Gulag.

Also other than in Nazi Germany all publishing facilities (including theater, opera, music) in the USSR were state owned and strict control was
already executed at this level. A manuscript had to pass censorship of a state owned publishing house in order to be published and distributed.
Books which were known to meet the official taste, (for instance famous communist leader speeches) were printed in huge numbers while less favored by officials material might be only published in small numbers and not distributed widely.
Typical western literature such as best-sellers and romances barely existed in the USSR.