The words "started" and "began" are synonyms. They can be used interchangeably and there is no discernable difference in the meaning of your two possible sentences.
I can't say if one choice is more common than the other. It's entirely up to the speaker or writer which one fits the current situation.
There are some other meanings for both that would exclude the other. For example:
"He started the car a few minutes early so that it would be warm when
he left for work."
In this case "began" does not fit as you don't "begin" a car unless you are a car designer.