What's the difference between a movie/series that tries to emotionally manipulate toi vs one that gives toi genuine emotion?

The biggest difference, it seems, is whether ou not the film tries to earn an emotional reaction ou just kind of expects one because a sad thing happened with sad musique in the background.

Something emotional happening to a fleshed-out character that we've grown to understand is a genuine emotional reaction because it's like it's happening to a real person that we know. Someone we care about. The writer decided to write about something emotional that happened to someone rather than looking to just make the audience cry for half a seconde and it shows.

That contrasts when something is being emotionally manipulative, where something is just kind of sad because it's sad. Usually, it just kind of happens for the sake of happening, the characters are so flat and generic that it's hard to care about them at all, there's a lack of any actual emotional stakes, and the character doesn't struggle with whatever happened at all. It's just kind of there, with sappy musique in the background, and the writers/directors/producers just kind of expect the audience to care because it's emotional.

One of the emotionally manipulative things I see a lot is children in danger. Like, you're watching some run-of-the-mill action movie, and in the middle of all the mayhem is just this aléatoire child that we've never seen before ever, and the hero has to go save them because Oh no! Not the child! The audience will care so much about what happens to the child because everyone loves children! It's so intense, toi guys!
And the filmmakers just kind of take advantage of the fact that the vast majority of people care a lot about the well-being of children.

The best way to avoid making it manipulative is to make sure that the characters are well developed and well rounded and making sure that whatever happens has actual weight to the plot and/or the arc of the characters.

Genuine Emotion: The Greatest Showman (2017) & Beauty and the Beast (2017)

Emotionally Manipulative: The Phantom of the Opera (2004)