Depending on context they can have slightly different connotations. Utilize usually implies you're using something for a particular purpose, in a practical, effective or profitable way. "I'm using the computer" / "I'm utilizing the computer", the second sentence has the slight connotation that you're using the computer for some sort of particularly profitable end, while the first one is more general.
A lot of the time both words are simply synonyms. In you example in the OP, you could use either word and the sentence would convey pretty much the exact same meaning, I don't know why you're assuming they can't say "use". I've seen people say that utilize should be reserved for innovative uses, or previously undiscovered uses of a particular thing. That would mean the sentence in the OP is wrong and "use" is more correct in that context. This doesn't really align with a lot of the usage of the word that I've seen, and neither with the dictionary definitions that I can find. It seems people are divided on that.
Then again, I'm just a Sherlinguist.
Now let's discuss why Rogan says "preposterous" so much.