If they want a particular console between the two that is what you should get. If they are indecisive or that won't work for whatever reason, if it's within your budget, I would suggest building a Windows gaming PC. With Big Picture in Steam, and modern video cards outputting HDMI you can use it as a living room device and it will be superior to a console in pretty much every way. You can use any controller you want with it, xbox, dualshock, Steam Controller, etc. and your back catalog of potential games is insanely huge. Graphics of a well built gaming PC will always be superior to console. The modern economics of games has made it such that almost every game you'll want to play will be coming to all three platforms, PC, xBox, and Playstation.
As a personal anecdote, my wife bought me a PS4 just last month for my birthday, because I really wanted to play The Last of Us. Now that I have a PS4, I went searching for console exclusives to play. I'm not finding more than a handful of games worth playing that aren't already on PC as well. And the PS4 is two years old, same as the xBone. Most games worth playing are simply on all three platforms now.
Another huge bonus of PC gaming is access to mods. You don't like the way a particular game mechanic plays, you can often find a mod to change it. I've found this incredibly helpful in increasing my enjoyment of a myriad of games. Since this is for your kids, using and understanding modding might also be a gateway for them to become more engaged with the technology. Many times it's not hard to write at least a simple mod of your own. Having the power to change something in game the way you want could be very interesting to them.