"I do not want to cry, but I am homesick."
Translation:En halua itkeä, mutta minulla on koti-ikävä.
Translate "vaan" in your head as "but rather," compared to "mutta" which means just "but."
Also, "vaan" is only used in negative sentence constructions: "Minä en juo kahvia vaan teetä." "I don't drink coffee, but rather tea"
Use this fantastic website here to learn all the ins-and-outs of the grammar you pick up through Duolingo: https://uusikielemme.fi/finnish-vocabulary/interesting-words/mutta-and-vaan-the-difference-between-conjunctions/