Citi Field is family friendlier, there are some childrens activities there - wiffle ball and a dunk tank. My 10 year old nephew wanted to do both, but we only let him do the dunk tank. Citi Field is also cheaper to get tickets at, which is family friendly.
I was in some pretty nice seats at Yankee Stadium, they were really well padded and a bit wider. The seats were the first row of the second section at field level, sweet! Here they would bring your food to you. Well you ordered it from your seats and it was brought to your seats, you just paid a 20% gratuity. Unfortunately, the menu wasn't that great so we didn't partake in this. The section in front of us had a better menu and got free bags of M&M's. So this is more convenient then anything that I saw at Citi Field but it's lack of selection made it not as useful. The atmosphere at Yankee Stadium was a little more serious and the environment was a little more hard core about baseball. And well there is something to be said for the Yankees players, they're all big names and with the right field being very generous to home runs games can be more exciting.
So if you're just looking for a fun baseball game then seeing the Mets and Citi Field will be fun, and you'll save a bit of money. If seeing the best players is your thing, then head to Yankee Stadium. But both stadiums will deliver a fun experience. Play ball...