Mewtwo has some good traits in competitive play, namely his high air speed, one of the longest wavedashes in the game, good range, great throws, an effective moonwalk, double jump canceling, one of the best projectiles in the game, and one of the best recoveries in the game. Unfortunately, Mewtwo has the unusual combination of a large hitbox with a light weight, making him very easy to KO. Additionally, another huge problem is that most of his kill moves are either slow or have terrible range. Combined with one of the worst SHFFLS in the game, arguably the worst tech ability in the game, being susceptible to being shield poked, and being hard countered by Fox, Falco, and Marth, three of the best characters in the game, it pretty much solidifies Mewtwo's place in competitive play. Additionally, one of the first Melee tier lists in 2002 placed Mewtwo all the way at the bottom, below even Pichu and Kirby, but tournament results by Taj, widely regarded as one of the best Mewtwo players in the world, helped move it out of the lower tiers, where it currently places 21st on the tier list. It doesn't see much competitive play at the moment, but top players such as Leffen and Mew2King occasionally use him in large tournaments.
So yeah, that's why Mewtwo isn't great in Melee. In Smash 4, Mewtwo is actually placed pretty high, and a lot of buffs Mewtwo was given in Ultimate is actually shaping up to top tier potential.