There were four total Ninja Turtle movies. The first was the most serious, dealing with Raph's anger issues resulting in his near death, Splinter's torture and disappearance, and the turtles subsequent abandonment. The second took a more cartoony approach, with new characters and stranger mutants. But Raph was still a bitch, Donatello went emo, and it ended with breakdance-fighting </Zoolander reference>.
But then we have the third. The turtles are living the good life now. They have a comfortable home in an abandoned undergound transit station. The Shredder is dead, and none of those bizarre mutants, aliens, and psychos like Leatherhead, Rat King, or Krang exist to bother them (small special effects budgets are great for keeping the world safe). But then April ruins it all by picking up a time travel scepter from an antique shop and propelling them all to feudal Japan. Samurais and stuff.
But you all know that story. That's not what makes this movie great. What makes it great is that for the first time we're seeing the turtles doing what they normally do. It's not an origin story. The characters and relationships are already established. The central conflict comes externally, not from within their group. This allows their camraderie to show through.
Raph makes a comment about how nice the environment is, and the others rib him for it. But he takes it in stride. Mike has to deal with his growing affection for one of the villagers, and whether or not to even go back. Raph finds his compassionate streak by taking care of one of the village's children. Leo and Don work on trying to get back while Mike and Raph may not be so keen to return to their time. In fact, they almost lose one during the transfer.
The fact that they're on an adventure, and they're all together, allows the turtles to act like they should. They crack jokes, they make fun of each other, they compliment each other when they make a good move (high foot!), they group together when things get tough. They act like heroes should.
That's the fundamental difference with the turtles and other contemporary superheroes. They actually liked what they did. Batman does what he does because he has to. He has to avenge his parents, or he'll go crazy. Wolverine does what he has to because he has no memory, and it's the only way his kind will ever be accepted (keep reaching for that rainbow!). Spawn, Spider-Man, Green Lantern, Superman, The Hulk. All of these guys take their responsibilities with aloof, maudlin grimness. Iron Man comes close to breaking the mold, because he enjoys a rich man's life, but he also has that obligation due to his past.
But the turtles are teenagers, although I never thought of them as such. It's hard to portray the real trials of adolescence when you're a large green shelled humanoid. You don't have school, or girls, or parties, or drugs, or sex. But I'm digressing.
They were just the turtles. They did the things that teenage mutant ninja turtles do. They eat pizza, they ninja fight some bad guys, they watch TV, they make jokes as they fight. It's what I would do if I had ninja powers. It's what I do when I play video games, killing zombies and Combine, making fun of the idiots trying to snipe me or yelling in sarcastic terror as the elevator goes out.
The fourth movie, TMNT (the CG one), was good too. If Ninja Turtles 3 did not exist, I would have no problem declaring this one as the best. It does well at exploring the darker side of the turtles as they grow up and begin to split apart from the bonds that made them so good in the first place--their neuroses, their hangups, their straining relationships in light of having no real goal to fight for anymore. It's a fond salute to the people like me who were so into them in the first place.
So, in conclusion, watch the movie again. You'll be glad you did, if you can get past the weird animatronic costumes.