Answer:
Antithesis?
Explanation:
An antithesis is when the author places two statements of contrasting meaning next to each other.
In this case, "the more things change" is literally the opposite of "the more they stay the same", but the author has placed them next to each other.
Another possible answer could be an "oxymoron", when the author uses contradictory statements together in a way that actually almost kind of makes sense. (Some good examples: alone together, deafening silence, organised chaos, etc).
In this case, "the more things change" and "the more they stay the same" are totally contradictory, but the author uses them together, implying that things can change but ALSO stay the same, which kind of sort of makes sense. Kind of.
Hope this helped! :)