This feels more like an English language than a personal finance question.
In a family-run business a founder or proprietor might enlist family members with varying qualifications (might have formal education/training or not) to help run the business. These people may or may not get paid.
In a professionally-run business, the owner(s)/shareholders hire executives/professionals/specialists to run the company and pay them for their skills/labor.
Note that a business can be both family owned and professionally run at the same time, there's no cut-and-dried distinction.