Honestly, they're often synonyms.
"Convert" implies converting TO or FROM something, example "Kinetic energy converts to heat energy."
Convert is most often used for converting formats (i.e. convert a Microsoft Word document to a PDF) or converting people to a new religion.
"Transform" implies a more dramatic and bigger change, and the word stands on its own (no need for TO or FROM).
E.g. "The desert transforms into an oasis after a rain" or "the caterpillar transformed into a butterfly."
It's as though convert is simply switching between two modes, but transform is to change more completely.
Hope that helps!