Program" and "programme" mean just the same. They are both usual words in English and they can replace one another in any context, as they are perfect synonyms.
Given the fact that they are both correctly spelled and have the exact same meanings, why are they still spelled differently? What makes them different and how do you know when you should use each? The key is the type of English you use: British or American.
Program vs. Programme
Besides your own preference, the only restriction - even though it is a subtle one - for using "program" and "programme" correctly, is given by the different spellings accepted and recommended by dictionaries for UK and US English.
British English, or UK English, recommends using "programme", whereas "program" is the form preferred by US English, or American/Australian English. Other than that, their meanings as nouns and verbs are completely identical. As nouns, "program" and "programme" are defined as TV shows, plans for events or activities, thin books or papers for shows or theatres telling about what will happen during the event, or as a set of instructions introduced in a computer. As verbs, they define the action of introducing a set of instructions in a machine.
When do we use "program"?
Use "program" in US English, especially in an official conversation. A simple trick to remember it is that Americans usually prefer shorter forms of words, to make it easier to pronounce and write.
When do we use "programme"?
In all sentences where you would use "program", but in UK English. British specialists will certainly advise you to use "programme" for more elegance and correctitude in your speech.
Even though these slight linguistic restrictions illustrated in all dictionaries recommend that you use "programme" in UK English and "program" in US English, in the end it's a matter of personal choice and of which version is more comfortable for you, especially in daily, informal conversations. According to the fact that they mean exactly the same and are both correct, as verbs and nouns as well, it is your own preference the one that, ultimately, should decide whether you spell "program" or "programme".