The difference between programme and program
TLDR: For US English, just use program. For everyone else, programme is non-computer-related behaviours or events.
Essentially, when using the word outside the realm of computer programs, the difference is in location. While those using British English use programme, our American and Australian friends would use program.
programme (noun)—a planned series of events, performances or activities; a booklet containing information on a performance; or an item broadcast on TV or radio
Roger looked in the magazine and drew out the monthly talks and seminars programme.
The Solar Power programme is progressing well.
I never miss a single David Attenborough programme!
program (noun)—computer software that controls the operation of a computer or other mechanical system; an application or app
My husband worked on a computer program recently.
The word processor I use is a faulty program. I hate it.
programme (verb)—caused to behave in a certain way; arrange within a schedule
Many birds are programmed to migrate.
The next school music night is programmed for next term.
program (verb)—to provide instructions for a computer or mechanical system; inputting instructions into a computer or mechanical system
To program all day long might be fun for some, but not for me.
To get going, simply program in your destination!
As you can see, anything to do with writing computer software uses the American spelling, while all other uses of the word depends upon the locality of the speaker.