You can use them interchangeably most of the time, except fora few cases where you can only use start.
The movie starts at 7:00. The movie begins at 7:00
It's starting to rain. It's beginning to rain.
Start a journey - If we start at 6:00 we'll be there by 8:00.
Start working (for machines) - The car won't start.
Make a machine start - How do you start the tractor?