It's important to me, because it's always a plus to have a large catalogue of games.
One of the reasons why I like the PS Vita is because it's digitally backwards compatible with PSP games. And also has PS1 games.
Honestly, Sony Computer Entertainment always did a fantastic job with backwards compatibility and emulation-based BC. (On PS2 [PS1 games], PS3 [PS2, PS1], PSP [PS1], PS Vita [PSP, PS1]).
It's a shame that PS4 has none of that. :l PS3 games would be extremely hard to emulate of course though, so that's understandable. But hypothetically speaking, how amazing would PS4's catalogue of games be, if it could play PS1, PS2, PS3 and PSP games and of course PS4 games?
I really don't like that they seem to propose PS Now is an alternative to backwards compatibility. It's nice as an extra option, but it isn't backwards compatibility, it's a seperate subscription service that have no ties with games you already own.
No backwards compatibility isn't a dealbreaker, but it's certainly always a big plus. Of course if I had a PS4, I'd mostly play PS4 games on it, but still compatibility with games you already have is always nice since it doesn't render the games from previous gen (PS3 for example) you own worthless if you don't have your previous gen system anymore.
And this is also a reason why PC is one of the best platforms for video games, because games pretty much get preserved forever regardless of when they came out. The catalogue of games spans decades and is immensily large ultimately. There's no segregation by platform generations like on consoles/handhelds if they said console/handhelds aren't BC.
It's a shame that PS3 and PS4 have such a discrepenacy in hardware architecture, because it would be great to have a system that could essentially play all PlayStation games ever released.