Can't put a finger on it...but it's that whole moth-to-flame effect.
Oh, I completely get that. When I watched him talking in his interview (it was one of Merv Griffin's) I kept pay attention to his eyes, because they're very expressive and show what he's thinking---which may not have anything to do with what he's actually saying. One moment he's playing "Richard Burton", another moment something would really reach him and the true man inside showed through. He had a mind like few I've ever come in contact with, and the ability to express his thoughts most articulately, never entirely unlaced with a sardonic humor, and of course I'm the world's biggest sucker for that combination. I get the feeling he once had pretty high ideals, and like many idealists, became disillusioned with the world and with themselves.
And when all this comes so attractively packaged, who can resist? He wasn't pretty, he had the kind of rugged looks I prefer.
You and I do have similar tastes, especially in actors and I think it means we have excellent taste, modestly speaking!
Here are the Burton films I have seen, which really aren't that many:
My Cousin Rachel
The Robe
Cleopatra
Look Back in Anger (bits and bobs---Brit kitchen sink dramas take a lot to keep me still, but for him I will do it)
Brief Encounter (this one I think was poorly directed---instead of a remake of the beautiful Noel Coward original, the director seemed only to go for the sordid aspects)
Where Eagles Dare
The V.I.P.s
Who's Afraid of Virginia Woolfe?
The Sandpiper (I like this one, believe it or not)
Becket (like you, only in fragments)
Ann of the Thousand Days
The Taming of the Shrew (excellent)
The Bramble Bush (I remember little of it)
The Rains of Ranchipur (only the last half hour)
The one I am most intrigued about seeing is Ice Palace, which was given a glowing report in another forum.
One of those films where his character is IN THE PROCESS of making a big mistake. 
