Belkar illustrates the classic DnD problem of the Mixed-Alignment Party. When most of the party is Good or Neutral, even a single evil (in particular, Chaotic Evil, or, as is sometimes known, Chaotic Stupid) character can destroy party unity over moral issues.
When you're killing a bunch of maurading monsters in a dungeon, it's not so much of an issue. There's no real problem with motivations since everyone regardless of alignment is on the same track; kill the monsters.
However, if the player of the CE character decides he hasn't been roleplaying his alignment well enough, or the Good players do something solely for the betterment of the world, then there are going to be serious issues.
In the first case, if the Chaotic Evil character decides, say, to resell libererated slaves to the highest bidder, or use villagers as walking health potions via a Vampiric Weapon (which actually happened in my game), then naturally the Good characters would want to stop such behavior. Conflict ensues.
In the second case, the CE character is going to feel he lacks motivation for the adventure, as rightly he/she should based off his/her alignment, then that player feels obligated to either walk out on the adventure or royally mess everything up at the first possible opportunity, or simply kill random people. Conflict ensues.
In terms of Belkar, he has been immensely useful in dungeon crawls and pitched battles- what DnD players call "Kick in the Door" kind of adventures. When the story has become more about "roleplay" aspects of the game, the alignment difference is going to cause difficulties in the same way that it would in a real DnD game.
That it does, that it does. I tried running an evil campaign once...about halfway through, I had to just change them into good characters because they sucked at playing evil (basically wanted to kill everything in sight, or anything that even gave them a cross look). I used jeweled armor (greedy characters) that had an enchantment that made the wearer's alignment change to the opposite of what they currently were. The enchantment only worked one time per person. Naturally they each received a set of armor, and put them on at the same time. They had no idea what was about to happen.
But, the humor value...was priceless! I have several stories I could go on about, but then again...that's not what this thread is for. Actually...come to think of it, they had more hilarious stories than just the ones in the evil campaign.
*Tips top hat, adjusts monocle, pulls out cane and walks away* and yes, that is Mr. Peanut laying unconscious on the curb.