These words are unrelated, and both have very narrow usages, especially braze, which is very rare.
Braise is a French word—both a noun and a verb in English—that refers to cooking something in a little bit of fat and liquid, at a temperature lower than boiling:
I’m going to braise asparagus with butter and garlic for a side-dish.
Like many French words in English, braise has aristocratic connotations. It is a word that one should know if one attends fancy dinners, but not for lower-class eating.
Braze is a much rarer verb. It now refers to the practice of joining together two pieces of metal using a softer metal, heated up, as a sort of glue—an operation also known as soldering (verb/noun: solder). Braze previously had a different but similar meaning—putting brass on other pieces of metal—but that meaning is no longer used.