I carry both religously, even if I think I know the area well. Rarely does the compass get used. My maps are handy most of the time if for nothing other than personal comfort, knowing I can get a good overview of the area I'm in - especially when I wander off the correct path.
With a well-groomed, well-marked trail, especially one that you're familiar with or have researched at home carefully, maps may be of little use.
Maps are invaluable when (1) forced to take an alternate route for some reason [blowdown, trail closure, forest fire, landslide, swollen waterway, etc.] or (2) to find an emergency bailout route [where does this trail/road go?] or (3) for when I want to go exploring from my campsite. That side trail might lead to a really secluded lake where I can skinny-dip or get away from the unwashed hiker trash.
Strip maps may be fine for the trail, but they're useless for anything else (ok, maybe as fire tinder) because they don't show features more than a few yards either side of the trail. Be sure you get large enough scale maps for the area you're in. Two perfect examples - the Trails Illustrated maps from National Geographic and the for maps of a particular National Forest/BLM area/Wilderness area.