The main difference is that foam can be from anything (soap, beer, chemicals, etc) and lather is mainly about soap. Personally, I've never heard the word spume.

Foam is the general word. It means many small bubbles together, like on top of a cappuccino or on the surface of the sea. Spume is a more literary or poetic word. It usually means the white foam on the sea. You will not hear it in daily life. I wouldn't even try using it with native speakers unless you are speaking to an English literature professor. Lather is foam made from soap. For example, when you wash your hands with soap and water, you make a lather. So: foam is the common everyday word, lather is soap foam, and spume is sea foam, mostly in books or poems.