Well while there would undoubtedly be at least *some* differences from OTL for sure(i.e. no Nazis for one), it may not necessarily make the world *totally* unrecognizable.
In terms of music, I find M79's suggestion in particular to be rather fascinating; swing, gospel and jazz mixing together, as an alternative, and possibly one that ends up being a worthy challenger, to either of the two other genres above sounds like it'd be fun to write about. A blend of *hip-hop and *salsa would be interesting as well. However, though, I'd also like to point out, too, that the roots for both *Jazz and *Rock and Roll were already pretty much in place, and just needed some time to develop and eventually emerge. There could very well be some interesting divergence from OTL, that is true, I think; but they'd still be out there.
And for the matter of social issues, America may not be necessarily be more conservative than OTL, either. In fact, without either of the Red Scares it could very well actually be a bit leftier than OTL; both of these were significant contributors to the resurgences of more hardcore conservatism in their respective eras and without them, a lot of the fuel for those fires becomes unavailable. I would speculate that, realistically speaking, the Civil Rights Act, or it's equivalent, would probably be signed no later than 1970 at the very worst, and possibly by the late '40s, early '50s at the earliest. Feminism will probably also do just fine and without the "Reagan Revolution" and the national rise of the Religious Right(who'd probably be largely restricted to the South), is liable to make even more progress than in our world.
Public apathy about racism may perhaps be just a little worse than in OTL for a while, without the Nazis and all that, but that's far from inevitable and when the true tragedies of Jim Crow start being aired, it's definitely going to take a good hit either way(though then again, the hard right may be even more eager to embrace bigotry than they were IOTL), and more people are going to want action taken.
All in all, the equivalent of the Sixties era, whenever it occurs, might not quite produce quite so many uber-radical movements and such, but large-scale social change is definitely inevitable at some point(barring, perhaps, in a totalitarian dictatorship, like an equivalent of North Korea if there ends up being one).
Also, on the geopolitical front, Russia may have a somewhat better chance of surviving the Marxist uprisings, but even then, sadly, a collapse might still occur. Their best bet is for Kolchak or someone like him to step up when the shit *does* hit the fan(and it will in some fashion, at this point). The Tsar *might* not do something incredibly stupid, but I wouldn't bet on that too much, sadly. He will probably end up in Paris or London after the rubble's cleared, Leninists or no.
The Ottoman Empire is almost irrevocably doomed at this point and will take a miracle to save. And there's a possibility the Middle East might not be entirely recognizable, either(whether or not Israel still exists is debatable).
Germany, if they don't fuck anything up real bad, or suffer a *Depression, probably will remain a Great Power, perhaps doubly so if they create a union with Austria-Hungary at some point.
Japan may enter a militaristic phase for a while but it probably won't turn into OTL's horrorfest.
China's Empire is gone but they may still yet become a Great Power.