To me they both sound fine. Perhaps with 'the' there is a bit more emphasis on only giving one answer?

This is a case of a technically incorrect shorthand becoming commonly accepted and thus "correct" in practice. That is, "impressed you the most" is the strictly grammatically correct version but people use "impressed you most" you might as well just accept it too. Take these examples: [1]"Who gave you most?" [2]"What part gave you most headaches?" [3]"What would impress you most?" [4]"Which one do you like most?" [1] and [2] sound plain wrong. It sounds like the way a Russian would speak. [3] and [4] are probably commonly accepted, but might sound uneducated to the well trained ear. Strictly speaking, if you wanted to leave out "the" you would put "most" before the verb as in the following: "What most impressed you?" "What would most impress you?" "Which one is most to your liking?" This style might come off as somewhat old-fashioned, or even sound somewhat awkward to modern day speakers.

What impressed you the most ? This is the only correct answer. Using What impressed you most... needs a preposition phrase after it to make sense. Example, .. What impressed you most... about the movie, about Mr. Bob, about Tokyo.