I stayed with MS after the 360, as I prefer their exclusives. Do I regret it? Not in the slightest.
Sure, the Xbone is graphically weaker but since I never have both consoles playing the same game in front of me together, I'll never be able to tell the difference. The games look good enough to me, and that's all that matters. I'm also much happier with how MS roll out their updates compared with Sony, and we seem to be getting genuinely useful additions: The new OS is due imminently too. MS are a software company though, so I'd expect no less. Certainly XBL appears to be more resilient than the PSN too, going on specialist media reports.
I like how I can upgrade the storage by just plugging an external HDD in too, and not having to open the guts up. I prefer the controllers too, far more solid than the PS4 ones but, then again, if you're used to the Dualshock you'll prob not like the Xbone controller. Personal choice and all that.
The BC works really well, as it's a full blown emulator rather than trying to be clever with software. Admittedly I still have my 360 set up in the retro man cave so I don't tend to use the BC stuff that much, but it's nice that it's there.
Ultimately, they're both great consoles. PS4 was much better at the start, Xbone now an equal. Above all buy the console your mates have if you do a lot of CoD/FIFA online stuff, or buy it for the games you prefer. Halo and Forza or Uncharted and GT, no right or wrong answer. All I know is that I love my Xbone, and I'm still not bored of pausing films by talking to my TV!