Garbage and trash are American English. The British use the word rubbish.

As Douglas mentioned, "rubbish" is the standard word used in the U.K., both for describing house-hold waste and as a synonym for "nonsense" (i.e. if someone has said something that you believe to be completely untrue, you could say: "That's absolute rubbish!"). I find it hard to say "garbage" and "trash" without giving them an American twang!

they are just different words to describe waste. Refuse can be used as a noun with the same meaning too. Refuse tends to be used in the UK in formal notifications.

In the United States, they are near-synonyms. However, in some communities there actually is a subtle distinction that can be important, because you may be expected to "sort your garbage from your trash." "Garbage" can mean food waste that can rot, stink, attract rats, etc. while "trash" means dry waste such as office paper. I'm not completely sure about "rubbish," now that my attention is called to it I recognize it as British but I think it must be used regionally in the U.S. as well. When I hear that word in my head I don't hear it being said with a British accent. Yes, http://www.penceland.com/ne_dialect.html says it is U.S. New England regional.

Humans are productive when it comes to elaborate useless words.