The words are not synonyms because the mental pictures created in the listener's mind will be different. For anyone interested in this way of thinking about words / definitions, I encourage you to read "Louder than Words."
All of my definitions are from The AHD, except Rubbish, from Cambridge Advanced Learners.
Litter: A disorderly accumulation of objects or things carelessly discarded. "The streets were filled with litter after the parade." This sentence will produce a different mental picture in a native English speaker's mind than "The streets were filled with garbage after the parade." This should be a clue as the meaning of garbage...
Garbage: Food waste as from a kitchen / worthless or nonsensical material. Garbage usually smells bad. No one wants to root through a garbage can. People wouldn't mind as much going through the office waste basket...
Waste: (verb) To use, consume, or spend thoughtlessly or carelessly. Waste has an idea of leftovers or things not entirely consumed. An office waste basket would have paper, paper clips, receipts, and other things in it that are considered leftovers or weren't used because they were extraneous. This is why we say "industrial waste," and not "industrial garbage / litter." In 100 years, hopefully, the expression "industrial waste" will be considered an unfortunate thing of the past, in the same way as "the electric chair."
Rubbish: AHD--worthless material, Cambridge--waste material or things that are no longer wanted. I include the British definition because in BrE, rubbish carries a similar meaning as waste in AmE. However, I feel that regardless of the speaker's native form of English, if you were to ask them to draw pictures of rubbish or waste, there would be significant difference and similarities. Also, in BrE people say "That's a load of rubbish," or "Quit talking rubbish." In AmE people would say "That's a load of BS," or "Quit talking trash / BS." (If anyone has some better "conversions" I would welcome those.)
Trash: To throw away or discard / worthless material. Also Empty words, worthless ideas, worthless or offensive literary material. It would seem then that trash and waste have similar meanings in some cases, but in others not so much. Waste basket and trash basket might be considered similar. But we don;t say "Quit talking waste," or "You're being trashful."
Refuse: Items discarded as useless or worthless. I think a refuse pile is very different from a garbage heap. In my mind, I picture old washing machines, printers, metal parts, bike tires, computer cables, car parts, etc. Garbage heaps smell and have things that are generally of lesser value.
In no way is this a scientific ordering but, I think one could order the words in terms of "grossness factor," thus:
1) Garbage 2) Trash 3) Rubbish (US perspective) / Waste 4) Litter 5) Refuse
I think it would be an interesting experiment to ask AmE or BrE speakers to describe the contents of "Rubbish bin" "Trash can," or "garbage can," and "waste basket."
Because the words "rubbish and "trash" are also used in expressions such as "talk trash / rubbish," I feel that the mental images created are different for these words than for other words and may in fact produce confusion as to what should be imagined...context possibly sorts this out.
This is a topic that could easily fill 20 pages of discussion about all the different uses, mental pictures created, word history, change of word meaning through history (check out Corpus of Historical American English), and etymology. Because of this, I don't think it's possible to give a quick summary that tells all.