Program is American English and Programme is British English. Occasionally you will see programme in AmE for effect. Other than that, they are synonyms.

Just something to add to Jerry's answer: In British English, there is a difference: We use 'program' to refer specifically to computer programs, and 'programme' for all other uses of the word (such as a daily programme, a TV programme, and so on). American English uses 'program' for all meanings of the word.