Aska from the SNES version. There was even a petition online to make her a Canon Immigrant!
Casey Jones and April O'Neil in the Genesis version. The latter because she abandoned the Damsel in Distress role to be more of a fighter, like in the comics.
Vindicated by History: The SNES version. The NES version to a lesser extent (though the main strike against it is the game feeling a bit uninspired).
Unexpected Character: Some of the characters in the various rosters are relatively minor characters from the more well-known '87 cartoon (like Rat King and Chrome Dome) and then there's a few Teenage Mutant Ninja Turtles Adventures-exclusive characters ( Armaggon is an Arc Villain, Ray Fillet was a major hero and War was just a Villain of the Week), plus Karai who at the time was exclusive to the Mirage comics, while Aska, and Sisyphus canon foreigners created by Konami exclusively for these games. Basically if you were only a passing Turtles fan, most of the characters would be completely alien to you. Even a lot of hardcore fans would struggle to recognize War, whom is definitely the most minor character to make the roster.
YMMV tropes of the Genesis version:
Catharsis Factor: In the Cowabunga CollectionCompilation Re-release, Konami gives you the option of playing as the three boss characters: Triceraton, Krang's Android and Karai. All of them were usually unavailable through any means except hacking. Those bosses giving you headache? Well, time to put yourselves in their shoes and give them headaches (provided that you can get around the Perfect Play A.I.).
Game-Breaker: Triceraton has the most powerful grab in the game, biting his opponent for half their health, which you can repeat with proper timing until they drop dead. Aside from that, he is fairly tanky, and hits very hard.
Heartwarming in Hindsight: The Genesis version wouldn't be the last time that the '87 cartoon's take on April O'Neil became playable, as Teenage Mutant Ninja Turtles: Shredder's Revenge would eventually come about and include her (jumpsuit and all, in contrast to Tournament Fighters) to fight alongside the turtles. This goes double for her playable appearance in Nickelodeon All-Star Brawl, which actually has her sliding kick attack from this game as a part of her moveset.
Nightmare Fuel: Karai's Evil Laugh from when you get a Game Over. Seen here. The factor of scariness is due to its surprise element: You have no idea how many continues you have left in one playthrough (and the game being super hard as is, you will be defeated a lot of times). While you're hoping to just press start and continuing, you're suddenly given Karai's surprise Evil Laugh as a Jump Scare, and the game ends right there.
YMMV tropes of the SNES version:
Game-Breaker: There are a few moves that are a bit too good. Donatello's strong punch has instant recovery, deals tons of damage, stuns fast, fills up 40% of his special gauge when his life is on red, AND has very good range. The computer can counter this by magic reflexes, but against a human...
Leonardo's 'Millennium Wave' takes pretty much the whole screen, deals around 45% damage IF BLOCKED. But if it connects, it deals up to 90% more AND stuns the opponent.
Michaelangelo's Rising Thunder takes over 30% of life if it connects on the ground.
Karai. Considered by many to be one of the most broken characters ever made in fighting game history. Her diving rapid punch special not only leads into a blockstun infinite, but it charges up her super meter incredibly fast. She also has a one-button infinite.
Nightmare Fuel: The giant octopus that takes up most of the Pirate Ship stage. It's massive eyes follow the fighters across the screen, perhaps waiting to grab the loser once the hard work's been done.
Viewer Gender Confusion: It's pretty easy for players to think that Karai is male, given that she's tall, broad-shouldered, and has a voice bordering on androgynous.
YMMV tropes of the NES version:
Fan Wank: Is Hothead the same character as Chu Hsi from the Archie Teenage Mutant Ninja Turtles Adventures? Same background, as both are former human firemen possessed by a dragon spirit, and both are credited as being created by Ryan Brown, but even the TMNT Wiki treats them like separate entities and Tournament Fighters is listed as being his debut (though that wiki treats every incarnation of a character as a separate entity, even the Turtles themselves). It doesn't help that his art on the cover is red (matching the "Hothead" action figure) while his sprite coloring is yellow (matching Chu Hsi).
Game-Breaker: Raphael's biting grab in the NES version can be spammed to make getting through the game trivial.
Germans Love David Hasselhoff: The NES version is the rarest of the three versions, yet the most commonly available NES TMNT game in countries where bootleg consoles and games were common, such as India.
The factor of scariness is due to its surprise element: You have no idea how many continues you have left in one playthrough (and the game being super hard as is, you will be defeated a lot of times). While you're hoping to just press start and continuing, you're suddenly given Karai's surprise Evil Laugh as a Jump Scare, and the game ends right there.