Alone feels like there are no people around so it works better with animate objects, where as "only one" could be about thing. "I was alone with the cake." "There was only one slice left." I wouldn't say "The rock was alone." unless I wanted to imply that the rock looked lonely, or didn't have anyone to help it.

These words can all be used to express the same meaning. Here are a few short examples: Sara is alone. She is the only one. It is just her. I would be careful to pay attention to what is being talked about in a conversation when hearing the word 'just' because it can mean different things depending on what is being said. Alone and only always have the same meaning. :)