For people wondering what the big deal with BC is, its really really nice to have during the launch of a console when there are so few good games. My PS2 at launch played quite a few PS1 games while I waited for Twisted Metal Black, Tekken Tag, and so on.
Of course after a year or two BC is pretty much just icing on the cake by that time you should have a pretty good library of games built up to keep the stigma of "it has no games!" away.... (The Vita could have used BC in this exact situation but I agree with Sony ditching UMD so yeah no BC is a pain, worth it? I think so but maybe it wasn't?)
On the PS4's BC issue, it should be compatible with the PS1 and maybe the PS2 via software. PS1 emulation should support all kinds of cool filters like the OpenGL ones... (There are some really cool OpenGL cell shaded filters and black and white stencil filters and cartoon filters for the PS1.)
If they don't have a software PS1 emulator at least, thats just sad
Edit: To put it really in perspective how important BC is, I spend more time playing PS1 and PS2 games on my PS3 than I do playing PS3 games....
Edit 2: I think Sony could have upgraded Cell to one of the other versions added all the RAM upgraded the GPU, kept BC with the PS3 and PS1... But AMD probably offered them a good price VS IBM's Cell.