I think the answer to your questions is heavily dependent on what you want to do after you get there!
If you are trying to pick a city to visit, then there is really no comparison between Salzburg and Innsbruck. Salzburg would be the clear choice for me.
On the other hand, if you want a hub for excusions north, south, east and west, then I can argue for either one.
From Salzburg the area in all directions has attractions. The same is true of Innsbruck.
For example, from Innsburck you could easily drive to Garmisch, even Füssen for the castles. Or a little farther still, even to Munich, although I might be tempted to take the train to Munich because of parking difficulties in the central city.
Also from Innsbruck, the Ötztal with a glacier near Sölden and the Timmelsjoch at the head of the valley. Also there are some decent waterfalls and a recreation of a village that is based on the life and times of Ötzi, the Iceman whose frozen remains are at home in Bolzano (Bozen) on the other side of the border in Italy.
At that time of year, however, I don't see the attraction of being outside a lot. Days will be short and the weather for me at least would be very negative.
Given the time of year, I would try to be finished by Christmas day because many business shut for extended breaks at Christmas.
Also with it being Christmas, in the final analysis I would pick Salzburg.
