Have never requested Christmas or New Year's but we were successful with a week 7 exchange there.
Having only stayed at Summit Watch, I wouldn't say there are a lot of differences in what each place offers, in terms of pools, hot tubs, fitness facilities, but the biggest difference is location. Summit Watch is at the end of historic Main Street, which is the centre of all the action in Park City. Mountainside is at the main base of the mountain, so about a mile away from all the action.
The ski in/ski out aspect is not a big deal for our family as we did not ski each day at Park City Mountain Resort, which is the hill that these resorts are ski in/ski out for. We also enjoyed Deer Valley, the Canyons, Alta and Snowbird.
One thing to note about the ski access is that if one is staying at Summit Watch, you get on the hill via the Town Lift. You cross a pedestrian bridge over the street and it's a small lift in the center of a little plaza right in the middle of town. At the bottom of Mountainside, you're at the base of the mountain, right next to the Payday express lift and all the skier services. If someone in your party was going to take ski lessons or was a beginner skier, you'd be in a better position at the Mountainside. To ski back to Summit Watch at the end of the day, only blue runs will get you there.
You can rent equipment from each location, but for ski lessons you must head over to the mountain base next to Mountainside.