Technically, they mean basically the same thing. In actual usage, they are different,. 'Pseudonym' is used when referring to an author or artist that uses a different name than their real one. Many times, the author''s real name is known. It is not associated with illegal activity in this case. We tend to use 'assumed name' is used for something illegal. Thieves and criminals will use assumed names. Note that I can use pseudonym for illegal activities and assumed name for an author's false name - but in general everyday usages that is not the case.

They have the same meaning but you would only say 'pseudonym' if you were talking about a writer, a painter or an entertainer.

From the definition for 'assumed name'from internet search: a false name or pseudonym, especially one used by somebody doing something illegal