Begin & start

1) formality
There is usually little or no difference between begin and start:
I began/started teaching when I was 24.
If Sheila doesn't come soon, let's begin/start without her.
We generally prefer begin when we are using a more formal style. Compare:
We will begin the meeting with a message from the President.
Oh no! It's starting to rain.

2) cases where begin is not possible
Start (but not begin) is used to mean:
a) 'start a journey':
I think we ought to start at six, while the roads are empty.

b)'start working' (for machines):
The car won't start.
c) 'make (machines) start':
How do you start the washing machine?