"Do you want a slice or a bigger piece?"
Translation:Tu vrei o felie sau o bucată mai mare?
If it's about food, or any portion/fragment that has come off or has been cut, broken etc. from a larger solid, then you say "bucată" in Romanian. So we usually have: bucată=piece.
If it's about a part or spare part of a larger system/machine/machinery, then you say "piesă" in Romanian. Think nuts, bolts, gears and other such doodads. So we usually have: piesă=physical part (of a system). Note that you can also use "piesă" when referring to: a theatrical play (piesă de teatru), a piece of music (piesă muzicală), a piece of clothing (piesă de îmbrăcăminte), a museum piece (piesă de muzeu), a chess piece (piesă de șah), and a few more which I won't cover here.
The words ori and sau are synonyms, they both mean "or". You can use them interchangeably, so the sentence should have been accepted. Report it next time. Btw, I think you meant to write 'sau' instead of 'și' above. Also, it does not have anything to do with the consonant/vowel of the next word.
I omitted "tu" and used piesă and when I was rejected I checked the questuions because I wrongly thought that I should have used the pronoun. I turns ou that piece has two different words. However in a previous sentence , the one a man has many pieces in a green jar, I used bucați and was accepted and duo used piese. Why either was correct in this sentence and not in the slices you explanation not withstanding?