Both 또는 and 아니면 mean "or" but 아니면 literally means "if not" so it can be used for slightly different emphasis. I think 또는 is also used more frequently in writing.
In my experience 아니면 puts preference to the first object. So like if someone asked what you want to eat you could say 피자 아니면 고기 implying that you want pizza but if not (perhaps if that's not an option) meat is also acceptable. Using 또는 feels like equal preference is given to both options and is more of a neutral "or".