The tone is what changes the meaning. However, on paper, the meanings are slightly different:
1. What do you mean by that?
This is a specific question, to a specific person, about something specific they said. Specific. It can be asking about the tone someone used or their underlying meaning behind saying something. This can be a harmless question asking for more information, or a very aggressive question, challenging someone to stop hiding their real thoughts behind clever words.
2. What is meant by that?
This is a more indirect question and asks what the 'general' idea is about a situation. This is a harmless question that would be useful for an English learner when asking what an expression means in a language. For example,
"What is meant by the question, 'What do you mean by that?'"
For the first question, just be careful about the tone you use when asking it.