Why do some animes have different endings between the manga and the anime?

Animation studios often have to decide between waiting for the manga to finish or getting ahead and creating their own ending, and that's what generates the differences you see. When the anime catches up with the manga (because they keep publishing chapters while it's being animated), producers have to choose: pause everything and wait years, or just come up with a different conclusion. Plus, sometimes they're asked for a more commercial or different ending than the manga's because they think it'll attract more people, so they deliberately take different paths. That's why you get cases like Fullmetal Alchemist or Evangelion where the manga and anime are basically different stories in the final stretch.

Besides what was already mentioned, there's a factor that sometimes gets overlooked: producers also cancel series early for commercial or ratings reasons. If the anime isn't selling well enough, they just slap a quick ending on it even though the manga keeps going, so you end up with two completely different stories. It happened to me where I started watching a series I loved and then read the manga hoping to continue, but it was like watching two different animes, haha.

It's not always because of that, honestly a lot of times the problem is more basic: the manga is still being serialized when the anime has to start recording, so the producers have no choice but to make stuff up or follow the manga as closely as possible until they run out of material. What was mentioned about cancellation also exists, but it's more a symptom that the anime just didn't catch on with the audience like they expected.