2 x 6's are nice on tall walls because they tend to dry a bit straighter. As far as strength goes, I dont think its much of a factor. The wall strength should be in sheer, and that's controlled by the strength of the joint where the plywood is nailed to the face of the 2 x 4.
In the other direction, you will get a stiffer walls with the 2 x 6's , but then again, its not like the 2 x 4's are an issue. I did add an extra 2x4 around the door frame.
As others pointed out, the main reason is to allow wider spacing the the studs. That allows wider insulation with less thermal bridging and of course thicker insulation.
If you go 24" on center you have less studs to use for supports ( shelves cabinets etc). You also have wider spans on your sheet rock.
Those are not big issues.... For simplicity in insulating the 6" studs work the best. One interesting thing, is with the 10' walls in my shop, the heat loss from the walls was pretty significant.
I went 2X4's with 2" of foam on the inside. It seems to work really well but its a bit more work.
Bottom line? Either will work, and 2x6's seem to be a good choice. For really cold conditions a layer of foam sheeting without thermal breaks is worth considering.