I've lived on the East Coast, West Coast, and in CO. If you want snow IMO CO is the way to go. The snow is light and fluffy, and not wet. Stay in Breckenridge, or Winter Park, or Copper Mountain, or Vail. Just because you don't ski does not mean that ski resorts are out of the question, and in fact you may want to give your boys some lessons, and see if they like it.
All of the resorts I mention have a LOT of fun stuff going on for the non-skier. Live music at multiple bars/locations generally starting in mid afternoon, cool restaurants, sledding, ice skating, snowmobiling, Xcountry skiing, shopping, and lots of other stuff. Crested Butte is one of my favorite resort towns.
If you need more information, or advice this is certainly the right place to post for Colorado and feel free to PM. It's NOT cheap to vacation here, but not a lot more than most resort places either unless you need five start accommodations in Aspen! Speaking of Aspen, another great resort town, but likely the most expensive of the lot.
As for snow, we've had a very dry year to date. However that WILL change. It's suppose to get very cold and snow in Denver and the Mountains on Tues night~Thursday. Denver is NOT the place to see snow. Despite what most people think Denver has something like 250 days of sunshine, and generally gets snow a few times a year in any measure. The mountains however can see snow daily for weeks at a time.
If you are looking for lodging on the cheap there are some places like Fairplay, or outside of Baily for example that have Cabins that my run less than in places like Breck. There are also towns near resorts like Eagle (Vail) or Silverthrone (Breck/Keystone) that are going to be less expensive vs. staying right in the resort town.
Renting a car is a bit troublesome. A 4x4 is a LOT more expensive vs. a car, and generally not needed. The rental companies won't even guarantee snow tires, in fact most cars will not have them. A better option maybe to use the shuttle service to the resorts to/from the airport.
Good Luck with the decision.