“Tu” and “tú” are the same spelling, except for the accent. Are words like “tú” and “mí” pronounced differently from how they would be without the accent?
Yes and no.
They are pronounced the same in isolation. However, the words tu and mi lack prosodic stress, that is, like a very limited number of words in Spanish (prepositions, articles, object pronouns), within a sentence they will not be stressed. When you say en mi casa, for instance, you only stress the first a of casa: /en.mi'ka.sa/. The rule in Spanish for unstressed words is that they will be pronounced as if they are part of the next word that has prosodic stress (this is why you have to write enclitic pronouns as one word, to show they are pronounced with the previous word instead).
On the other hand, tú and mí like most Spanish words do have prosodic stress and will always receive a stress: a mí me gusta is /a'mi me'gus.ta/.
The effect goes a bit beyond just stress though, as stress influences things like vowel elision or reduction (tu amor will probably be pronounced /twa'mor/ but tú amabas will have a clear haitus /'tu a'ma.bas/).
I would say they are pronounced the same and that the difference in writing is to distinguish one word from the other. There might be a difference in stress as the person above wrote, but I can't hear it for itself. "Tú, madre,..." (You, mother,...) would sound different than "Tu madre" (your mom) to me because there is a comma there that makes you stop, implies that you are talking to someone by name and therefore has a tone like in "hey, mother" and not because there is an accent. Same goes for "No te preocupes por mí, mamá" (don't worry about me, mom) and "No the preocupes por mi mamá" (Don't worry about my mom). If you are a beginner in Spanish though, I wouldn't stress about it ;)