Both towns have charming historic architecture. I agree that the altarpiece is wonderful (and I am not at all religious); I enjoyed the Unterlinden Museum a lot. In my experience (second half of June 2017) both Colmar and Annecy get very heavy tourist traffic, though I don't remember really significant crowding inside the museum in Colmar. The sidewalks in the historic district of Annecy were particularly packed, though I walked along the lake for quite a while and saw very few people. I'm more tolerant of crowded sidewalks than the average tourist, but I've seen a few negative comments about how "touristy" Colmar is on this forum, and I found the historic center of Annecy to be worse. I think we just have fewer people who've been to Annecy recently. I say this not to discourage you (having enjoyed my visits to both places), but so you are mentally prepared to be one of many visitors.
One other warning: Annecy is near the Alps but not in them. If this turns out to be a (Northern Hemisphere) summer trip, be sure you have air-conditioned lodgings in whichever town you choose. It was miserably hot at the time of my visit to Annecy. You can take a look at actual, historical, day-by-day weather statistics on the website timeanddate.com. I see that I managed to arrive at the beginning of five 90F days; such luck I had--but then again, 2019 would have been worse.
Edited to add: As to which to visit, I preferred Colmar by a considerable margin. The historic area seemed larger, the museum was a substantial plus for me, and it wasn't so blasted hot. But Annecy has the lake, which would be a big positive factor for many visitors; I enjoyed the boat ride I took. They are very different towns, so I don't think visiting both would feel repetitive. The little Alsatian wine villages are nice; you'll need to research transportation options for those unless you decide to rent a car for a day or two. There are some buses, but visiting more than one of the small places on a single day will probably be a challenge.