Begin or Start

"Start" (verb) means "to begin doing something." 

"Begin" (verb) means "to start to happen or exist."

Differences

1.   "Start" has more definitions than "begin."

"Start" can mean "to cause something to happen." This includes causing a machine to work or creating a business etc.

• "I've been having trouble starting the car."

Not - "I've been having trouble beginning the car."

• "Things have been going well since we started the business."

Not - "Things have been going well since we began the business."

• "Start a movement to change the world."

Not - "Begin a movement to change the world."

However - "A movement began to change the world."

"Start" can also mean "to move your body suddenly because of fright or surprise."

• "He started at the sound of the doorbell."

Not - "He began at the sound of the doorbell."

2.   There are more expressions/idioms with "start."

Begin:

• "To begin with"

• "Can't even begin"

Start:

• "From start to finish"

• "For a start"

• "Start out"

• "Start over"

• "Start something off"

• "Start on something"

• "Start something up"

• "A fresh start"

• "A head start"

3.   "Begin" is considered more formal and polite.

Compare:

• "When do we begin?"

• "When do we start?"

4.   "Begin" is considered more gradual; "Start" is considered more sudden.

Compare:

• "The number of views began to increase when I uploaded more videos."

• "The number of views started to increase when I uploaded more videos."

• "I was beginning to fall asleep."

• "I was starting to fall asleep."