From my (scant) experience in the leveling process, I think it boils down to the following;
Scrappers deal more consistent damage across a tougher enemy, given the highest base melee scaling for damage plus critical hits.
Stalkers burst harder but aren't consistent in their output as they have less outright scaling, but they can crit more often and have assassin's strike.
Brutes will soak the hits better than the Scrapper or Stalker, and mechanically punch back harder, but it boils down to how much fury they have. It will be consistent damage, but it's never going to surpass the guys who need a little bit of lady luck on their side.
Tankers just eat hits for days and bomb large groups of enemies for consistent, albeit lower, damage values. They're really the pinnacle of "I can do this all day." In a one on one fight, the other three melee folks will probably finish the fight faster, but the tanker will finish the fight alive.