Won For... Good Will Hunting
Should Have Won For... One Hour Photo
There's a fairly reasonable suggestion that Robin Williams should have won Oscars for all of his nominated roles (The Fisher King, Good Morning Vietnam and Dead Poets Society being the ones he missed out on), but there's an even bigger case that he deserved wins for two unnominated performances.
He could have won for Insomnia, and nobody would have called injustice, but he absolutely SHOULD have won for his excellent, clawing performance as Sy The Photo Guy in One Hour Photo. It's a complete transformation of a performance, creepy and dangerous in its mundanity, and bubbling with something darker and more sinister.
Sy is the archetypal portrait of a killer whose neighbours would say he "kept himself to himself", and where there was huge potential to play him as a pantomime creation, Williams instead offered something more considered and more effective for it. He's just as wounded as Dr. Sean Maguire, but he's a victim of a monster inside himself, and the lack of passionate explosions marks him out as the superior creation.