To my knowledge, the characters all play exactly the same, and the "health" thing is just a crock of stupid, as they start off with 100HP and take the same amount of damage.
Captain America is the leader of The Avengers, and bizarrely, is Player 3 in the 4-Player version of the game. Given enhanced powers by a super soldier formula, he battled stereotypes of Nazis and the Japanese up until it became politically incorrect, so now he battles more generic evil with The Avengers and disses France.
Real name: Steve Rogers
Iron Man is the president of Stark Industries, a manufacturer of weapons, and uses his numerous robot suits to fight villainy and frequently has drinking problems. Also the only guy with facial hair. Is Player 2 in the 4-Player version.
Real name: Tony Stark
Vision the adventuring android is pretty wacky, as apparently he was created after the original Human Torch (who was a robot) was split into two identities by Ultron, and joined The Avengers. Going by a different look from his usual green and red colour scheme, he's Player 1 in the 4-Player version for whatever reason.
Real name: n/a
Hawkeye was originally a comrade to the Black Widow, and committed numerous crimes until he learned the error of his ways and joined The Avengers. A bit of a dickhead, he wields a mean bow and arrow and is Player 4 in the 4-Player version.
Real name: Clint Barton
Job: Security guard