Personally I think both have their benefits, so I use both. However I use Endomondo to track and record the rides, and then upload to Strava when I'm back for the segments and such as I think it has more focus on 'personal bests' although I rarely ride through an area thinking there's a segment here, speed up!
When I tried Strava I wasn't to impressed with the map-view, I wanted a bigger view of the map, and that might be down to my own lack of fiddling with the app to get it how I wanted, but I just found Endomondo easier to 'start and go' and check if I got lost.
