In a modern world, the Teenage Mutant Ninja Turtles cartoon of the 80s would have been considered a putrid pile of crap manufactured to sell toys to kids and had nothing to do with the source material. The key phrase here is source material, as in, anything less would be considered an embarrassment. Of course, that's not the case for Ninja Turtles. The 80's cartoon has become the codifying piece of the franchise; when people think Ninja Turtles, they think of this show. Somehow, they got away with turning a gritty black-and-white comic into...whatever THAT was. But if anyone else tries to pull this shit, they get rightfully slammed.
Be it Sonic Underground or Loonatics Unleashed (which was especially infamous since they retooled the show in season two), if it doesn't follow the source material, it's not worth a damn. Yet, The Spectacular Spider-Man had almost nothing to do with the comic of the same name and it was the best thing to happen to web-head since the 90's series. When people talk about The Office, you bet your ass they're talking about the American version. And, for a real mic dropper, the definitive example, Buffy the Vampire Slayer had zilch to do with the movie, and no one gives a shit.
In fact a lot of the more popular adaptions, even if they don't follow the source material, are still considered pretty good, if not better. That's because they're great shows on their own terms. One needs only to compare The Spectacular Spider-Man to Ultimate Spider-Man to see why. It's doesn't matter how faithful you are, if the show sucks, the show sucks. Back to Ninja Turtles, only the 4Kids adaption was faithful to the original comic (at least for five seasons...). Yet, it's considered as good as the 80's cartoon, because it was good show in its own right (again, for five seasons...).
Contrast this, with The Next Mutation and the recent live-action movie. In the modern world, we consider them putrid piles of crap that had nothing to do with the source material, because that's the goddamn truth. When all is said and done, it doesn't matter whether or not you stick to the source material. What matters is if the adaption, on it's own terms, is good. That's why remakes are so notorious. If it doesn't follow the original story, it's considered inferior. But even if they do follow the original faithfully, they might get complaints of being too faithful, they'll be considered a cardboard cutouts that offer nothing new.
So, how did Teenage Mutant Ninja Turtles get away with it? Long answer: they took the original concept and did it in an entirely new style that struck a cord, not just with kids, but with those who read the actual comics. Even if they don't consider it better than source material, they can see why other people love it.
Short answer: It was awesome. Period.