"She is very happy with her new coffee maker."
Translation:Elle est très contente de sa nouvelle cafetière.
Prepositions are not consistently translatable one-for-one across languages, not even languages as similar as English and French. We just have to memorize them.
Auxiliary prepositions after verbs, in particular seem to vary a lot.
In this case, "happy with" is always "heureux/se de" or "content(e) de".
"Contente" and "heureuse" are both accepted here. Check your full answer because Duolingo sometimes gives you the wrong error message. For example, it's not "avec" but "de".
As for the difference, they are generally interchangeable. "Contente" is more of a fleeting feeling, while "heureuse" is long-lasting and specific. But they are used for either meaning nonetheless and no one bats an eye.