Most of the kid-friendly games are cross-platform so that they can hit as broad a market as possible, which is a good thing for you. (LEGO games, Disney Infinity, Skylanders, etc)
Having said that, the XB1 has Project Spark, which is sort of a more freeform "build and play" type thing, Zoo Tycoon, Screamride, as well as Rare Replay which is a ton of classic family friendly games (other than Conker).
The PS4 has Little Big Planet 3 for make-and-play as well as Tearaway. There's also Knack, but that's probably too combat focused and difficult for younger children.
Minecraft is cross platform, but of course Microsoft owns that now so it's hard to know if they'll ever do exclusive extensions or extra work on XB1 (I'd guess probably not, since they want to satisfy their whole audience).
If you care about racing games, the Xbox definitely has more and higher quality ones right now (Forza 5/6, Forza Horizon 2) as well as all the cross-platform ones. The PS4 has Driveclub which came out poor but has been patched since (although still not a great game). They'll also eventually get a new Gran Tourismo, but not until late next year at the earliest. There are of course also cross platform Need for Speed, Project CARS, The Crew, etc.
I think sports are pretty much even across the board, except the PS4 has an exclusive baseball game that Sony makes (MLB I think?) if you like those.
There haven't been a ton of Kinect games for XB1. It does work better than on the 360 (better angles, more fidelity, better microphones), but I think for most early adopters the thrill had worn off so it's kind of gone by the wayside. There are a few games that need it, like Disney's Fantasia, Dance Central, Kinect Sports Rivals and there are some that have extra functionality for Kinect but don't require it like Zoo Tycoon and Project Spark. If your family really liked Kinect (or might now) it's probably worth it for kids, but don't expect a ton of AAA games in the future (there are some indie games that use it, though).
If you have a lot of 360 games, there's a chance some of them might be part of the backwards compatible push this fall or later, if that matters to you.
FWIW, I have both but I prefer my Xbox One, so that may color my opinion.
You probably can't go wrong with either of them. In general the PS4 is leading the market, but that also means you may be able to get a better deal on XB1. They're both selling better than the consoles of the previous gen, so neither is going to fail. I'd say try to make a list of games interest you and your family most and then go for that. It's a bit of a personal choice, so I think you'll be happy with whatever has the games you want right now and whichever ecosystem you prefer.