They are completely different. A "premise" is part of a chain of logic. "On the premise that X" means "If X is true." "On the premise that any exposure to my target language will help me learn it, I have set my computer's language to Portuguese." A "pretext" is a kind of lie. If someone says they are doing something for good reason X, but they are really doing it for bad reason Y, then X is a pretext. For example, "The child told his parent that he had a headache, but he didn't really have one. It was just a pretext for staying home from school."

"On the premise" is used as the basis of a theory, real or assumed. Example: The stranger in your house was arrested on the premise that he was a thief. On the premise also can imply a location. Example: I am on the premise ( a location already determined). "On the pretext of" is hardly used but used more as "under the pretext of" and is a false reason given so that you can hide the real reason. Example: He came under the pretext of an invited guest, but in fact stole all the family's belongings. I hope this helps!