I don't know about Diablo 3, but Diablo 2 I did play in several different ways, and the results were interesting.
first time, I picked a Necromancer. I thought I could benefit from the horde mechanic and just swarm the enemy with skeletons, but I found it hard to control and unable to defend against wizards that'd just bypass the horde altogether with ground attacks or meteor strikes. So, I dropped it and started over.
second time, I picked a Paladin. Went into it playing as a medium warrior, with sword and bow, not really using the Paladin's core power which are his auras. I managed to beat Diablo by stacking up on potions and basically out-maneuvering him until I could land a couple of arrows. Once I beat it, I didn't really "feel" it... I was relieved for beating it, but didn't feel satisfied. We didn't really "fight", all I did was dodge and shoot out of screen.
third time, I installed Lord Of Destruction and picked up a Druid. Wasn't really convinced about the shapeshifting thing, so I dropped it and picked the Assassin.... again, wasn't really digging the 1-2 mechanic for finishing enemies, so I dropped it as well.
then I went and really thought about it, and came up with a controls structure that'd really benefit a Paladin and it's auras, so I changed the controls, and ended up with a sort of gearbox structure with offense/neutral/defense stances.
fourth time: took the Paladin again, with that structure in mind.... and boy, I was DANCING my way through. It was totally intuitive: buff up, switch to defense, tank, rush in, strike, switch to heal, etc.
I picked up the Amazon companion at the first stage, and carried her all the way to Baal. SHE killed Diablo, while all I did was tank hits and heal both of us (you know... as a Paladin would...). And for Baal, we both died several times, until I remembered I had that one thawing potion I picked up somewhere down the line and totally forgot about it, stood my ground in front of him, and using that gearbox mechanic I was changing between healing, defense, attack and buffs in between his attack runs. He froze me, I switched to reflect damage, he froze himself, I drank the thawing potion, switched to shield bash, stunned him, and hammered him down with Blessed Hammer.
That one I'll remember, because everything went according to plan: he had 1 HP and that 1 HP would cost me everything I had, because I had nothing against his freeze except that one thawing potion.