We take that route quite frequently as our son lives in the Phoenix area. 277 is a relatively flat road and if it's snowing there it will probably be snowing in Flagstaff as well. 2 lane until you reach the top of the mogollon rim then it is 4 lane most of the way to Phoenix area.
Down side is that you travel over a number of small ridges. Some of them steep enough to have slow you quite a bit. Up side, you don't have to drive around the phoenix loop. The loop is fast moving and heavily traveled highway.
Time wise you will probably find the trip about the same, oh wait, I see you're driving a Dodge. Better take the Flagstaff route.