Of all the random things:
Cyberpunk won't be as big as it was in the early 90's. Part of the appeal of it appeared to be a response to the sudden removal of the "certainty" that the world would end in nuclear fire. So instead of a four minute warning, blinding light and a fire storm, we had megacorps, Street Soldiers, Net Runners and ripper docs, either way, the world would be destroyed. OTL, now, we can see from films made by people of my age, a rush of Zombies, another allegory to deal with having such a grim certainty ripped away, (in the best possible way).
Music, possibly not too much of a change, Grunge would be darker, so more of an Alice In Chains sound than Blind Melon, and Industrial would keep it's dark sound. Gothic Rock would last a little longer. The videos would have more of a post apocalyptical feel to them, Kind of like
This.
I think Glam Rock would survive a bit longer than OTL, it won't be killed off by Grunge this time around. It's message of hedonism will still be an escape from the grim reality, so expect that to be even more outrageous than it was.
Comics would continue their WWIII allegories, Battle and 2000AD especially, more from Judge Dredd, Strontium Dog and Rogue Trooper.
Television would probably have more satire and Spitting Image would last longer, although series like Soldier Soldier, Our Girl, The Professionals would last longer, and Spooks would probably start a little sooner, but possibly with more Anti-Soviet storylines.
Books like OTL's Bravo Two Zero and The One That Got Away would be more popular and probably less obstructed by the MOD.
That's all that springs to mind so far.