Very versatile, easy to bring into UU games (although it's got a rival in Muk), and it can Explode. In OU, you will find it getting hit hard by almost everything, and not able to get past Recover-using Psychics, but at least you can blow it up there.
Weezing is like Muk: somewhat usable in OU, but mostly suited for UU. It excels in Defense, but is hindered by the Earthquake weakness. In OU, try to use Explosion or run when dealing with Chansey and Alakazam, and scout for paralysis, critical hits, or whatever luck you may get against Starmie switches. Fire Blast hurts Exeggutor only slightly less than Starmie's Blizzard, which is quite a lot.
In UU, Weezing can switch into Grass-types pretty easily. Use your prediction skills and its versatility to hurt your opponent on switches, and when Weezing is through, Explode.
Haze is there, but it's really iffy. There's very few things that use stat-up moves, and the few that do should just be dealt with by using direct damage. Haze also cures your opponent's status afflictions, which is really annoying if you managed to sleep or freeze the right Pokémon. You'll generally want to stay away from it.
Checks and Counters
Just how many things does Gengar screw up completely? Add Weezing to the list. Psychic Pokémon such as Exeggutor, Alakazam, and Hypno stop Weezing—they will shrug off its attacks and 2HKO it with those strong STAB-boosted Psychics, but Weezing can Explode on them. Also, Ground-types kill Weezing, especially Rhydon and Golem, who will resist just about anything Weezing will try to use.
In UU, Weezing is a little harder to counter. Bring in powerful special attackers on Weezing's special moves, like Electabuzz or Psychic Wigglytuff. Earthquake dents it badly, and Psychic should nearly always 2HKO it. Marowak, Onix, and other Ground-types should do a really good job as well.