“Begin” is usually used in a formal language and it often suggests a beginning of a series of events. “Start” is more common in less formal use and spoken language, for example:

We’ll need to start at 7.00.

“Start” comes in handy when we refer to turning on a device or a machine, for example:

The car won’t start.

We also use it to describe the beginning of a specific phenomenon.

Who do you think started the fire?