Begin and start

Meaning / formality 

Begin and start can both be used with the same meaning:
I began / started teaching when I was 25.
If Matt doesn't come soon, let's begin / start without him.

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!

Cases where begin is NOT possible

Start (but not begin) is used to mean:

a. "start a journey"
I think we should start at 5am, while the roads are empty.

b. "start working" (for machines)
The car won't start.

c. "make something start"
How do you start the washing machine?