Of all the Canadian province and territory highpoints, New Brunswick's Mount Carleton is the only one that is a traditional summit reached by a normal hiking trail. All the others are major mountaineering expeditions (AB, BC, YT, NT, NU, QC, NL), nondescript flat spots (SK, PEI), remote wilderness/bushwhack destinations (ON, NS), or a drive-up (MB). If you want a provincial high point where you drive to the trailhead, hike uphill for a few hours in a pleasant forest, and enjoy nice views from a rocky summit, Mount Carleton is your only option.
Carleton is part of the Appalachian Mountains, which are a very indistinct grouping of peaks in New Brunswick. And despite its relatively low elevation compared to peaks in nearby Maine, Carleton is clearly the dominant summit in the area, with an impressive 2000 feet of prominence.
Hikers who climb Carleton by the normal Mount Carleton hiking trail should, if time allows, hike north a ways to Mount Sagamook. This slightly lower peak has a north face that is the most impressive mountain wall in all of New Brunswick.