There's no difference between saying 'Put it in the rubbish bin' and 'Put it in the bin'. There are other types of bins, such as dustbins (large bins out of doors, usually for household waste) and wastepaper bins (smaller ones indoors), also recycling bins, and so on, but in most cases, the word 'bin' is sufficient. If you say 'Throw it in the bin', it's clear what you mean. There is no need to say 'rubbish bin', because it is obvious what you mean.