It's a control issue.
Handicaps are created by a tournament director or by a computer system, and can be done either at the spur of the moment or over a long course of time. Handicaps can be skewed based on prejudgements, personal biases, jabber and whining by other players, length of time and consistency, or math calculations that attempt to quantify something personal that changes constantly. The player has very little if any immediate control over his own handicap.
Weight is determined by the two players, by themselves, that day, based on how they feel about themselves and each other, right then. Weight can be skewed by biases, pride, confidence, fear, desperation... but it is a decision made by both players right then and there. Weight can also be changed quickly, during the course of a night, with a little more bickering. Can't do that with a handicap.