I just purchased a 1TB Xbox One a few weeks ago. I typically purchase most major consoles per generation (although I'd tend to buy for Xbox if it wasn't exclusive), but wasn't impressed enough with any of the new generation until Fallout 4 pushed me over the edge. Here's why:
* I'm an Xbox gold member, and they recently started offering me free Xbox One games, so I already had a library building up
* Xbox 360 compatibility means I may be able to ditch my 360 sooner
* I prefer the Xbox style controller
That's pretty much it. I've also been purchasing all my games digitally, which means never having to swap discs ever again, although it meant a week straight of downloading on my poor DSL connection. I'm guessing once I fill the internal drive up, I'll then have to start actively managing the remaining space, so there's that downside, but I really like going disc-less so far. I never traded or sold used games, so this is a win for me. My understanding is that you can purchase all games digitally on both platforms. If your kids like to swap / borrow games with friends, then perhaps discs would be better for you.
I'll eventually get a PS4 so I can play some of the exclusive JRPGs that always seem to land there, but for the time being, I'm fine with the Xbox One. The two consoles are, to be honest, *very* similar in specs, with the slight edge going to PS4 - so slight, in fact, you generally need side-by-side comparisons to tell the difference.
MS has the better online experience, from what I've seen (considering I don't have a PS4 yet, it's been a limited view), and for $60 a year you get a number of free games every month, though I understand PS has something similar. More than likely your credentials and credit cards will be safer with MS, as they seem to have a better handle on network security, as demonstrated by repeated Sony breaches, but hopefully they've learned their lesson.
Overall advice: get the one you like, especially if it has an exclusive you really want. Most games will be available for both. I think your kids will be happy either way. If you really dislike MS that much, maybe go for PS4. But really... it's sort of a toss-up.