With kids, I'd go to AK.
We're also arriving on Christmas—flights were TOOOOO expensive on 12/24, even with our points.
Neither park will feel overwhelmingly crowded. They both absorb crowds well, unless it's NYE.
EPCOT is great during the holidays. In past years, they've had the holiday offerings around the World Showcase (a very, very scaled down Food & Wine—assuming they'll do it again); there's plenty to learn about the different countries and their traditions; you can go to the Candlelight Processional; there's a bunch of different places to eat—whether you celebrate Christmas with Italian food, a steak, French... whatever; and, it's close to Beach Club, Boardwalk, Yacht Club and Swan & Dolphin, which all are beautifully decorated and have their own assortment of food places.
In the past, AK has been the least crowded park on Christmas, but I'm sure that'll change this year with Pandora. If I'm a betting man, I'd say DHS is the least crowded, although we expected that 2 years ago and were pleasantly surprised (the new Star Wars movie played a big role in that.) AK obviously has more rides and a few more things to see and do.
There will be more to do with the kids at AK, but EPCOT definitely has more of a holiday feel to it. It'll come down to: would you rather go on rides or be in the holiday spirit on Christmas?