3DS seems like the no-brainer to me. I have both and am honestly surprised to see so many Vita votes, but then I am on GAF.
99% BC with the DS library. Vita has only digital BC with certain titles (though there are many and many of them are good.)
Ultimately, it depends on your tastes. Just go to GameFAQs and take a look at the libraries in your preferred genres.
For me, I like JRPGs. I have several on the 3DS, with more coming this year, and Persona 4 Golden on the Vita. With FFX-2 coming sometime in the future, and I'm not even sure I want to buy it.
I'd wait until we see how well remote play works between the PS4 and Vita before letting that be a deciding factor. Less buttons than the DS4 means programming inputs to the touch screen or rear touch pad means controls will often be inconvenient, which combined with a laggy experience like PS3 remote play would only be appetizing for the Sony fans.
Vita also has its ridiculously priced memory cards to deal with, compared to 3DS using normal SD cards.
Vita is a great machine to play PS1 games on. They look better on the small screen than they do on an HD television. It's very serviceable to play PSP games on, if you don't mind buying all PSP games digitally. The PSN account works nicely and PS1 games and saves are transferable between a PS3 and Vita. Vita saves are not as convenient to transfer. PS+ does offer a lot of content for your Vita, there's no denying, and you're being forced to buy it anyway if you're getting a PS4 and like online multiplayer.
tl;dr: Follow the games. IMO, 3DS has more of them and more coming. Others will tell you that Vita will be full of great games in the future, but they've been saying that for going on two years now. Depends on your taste.