In practice there is almost no difference. Some phrases work better with one word than the other. Some examples: Start your engine, Start the washing machine, NEVR, begin your engine. But ,"you may now begin the test", OR "you may now start the test". Either is OK. Begin seems a bit more formal to me. Mostly it is a matter of usage, and simply listening to a lot of English speakers will help you know when to use start or begin.

If you are using start as a noun then they are not interchangeable. For example you could say "Let's hurry up or we will miss the start of the game", but you couldn't say "Let's hurry up or we'll miss the begin of the game". In that instance you would have to use "beginning". But otherwise I think they are interchangeable. For example "let's start the game" / "let's begin the game" are the same.