There is no difference between these two, except begin is more formal than start.
When did you start working here? When did you begin working here?
It is time to start the class/begin the class.
Two situations that you cannot use the world begin :
1. To tell about a journey
-I started to walk to the beach at 8am .
2. To talk about machines
- I can not start the washing machine.
- I don’t know how to start the photocopier.