The majority of folks hike Moosilauke from the Ravine Lodge or along the AT (Beaver Brook or Glencliff). The Carriage Road got more end-to-end traffic back when the Moosilauke Inn was in existence at the trailhead but now there's not as much traffic there. The section between the Snapper junction and the summit gets lots of use, but as it's more of a forest road than a trail people generally use one of the other approaches.
In winter it's very popular to go up from the bottom for skiers, but also for snowmobilers.