If I can, what's the legal difference between a business that is forced to not discriminate when hiring someone and a person hiring somebody to perform a service for him?
Title VII of the Civil Rights Act of 1964 regulates employers and employees. The prohibition on employer practices says
it shall be an unlawful employment practice for an employer -
(1) to fail or refuse to hire or to discharge any individual, or otherwise to discriminate against any individual with respect to his compensation, terms, conditions, or privileges of employment, because of such individual’s race, color, religion, sex, or national origin; or
(2) to limit, segregate, or classify his employees or applicants for employment in any way which would deprive or tend to deprive any individual of employment opportunities or otherwise adversely affect his status as an employee, because of such individual’s race, color, religion, sex, or national origin.
You are not an employer, which is
a person engaged in an industry affecting commerce who has fifteen or more employees for each working day in each of twenty or more calendar weeks in the current or preceding calendar year, and any agent of such a person
For completeness, an employee is
an individual employed by an employer, except that the term “employee” shall not include any person elected to public office in any State or political subdivision of any State by the qualified voters thereof, or any person chosen by such officer to be on such officer’s personal staff, or an appointee on the policy making level or an immediate adviser with respect to the exercise of the constitutional or legal powers of the office. The exemption set forth in the preceding sentence shall not include employees subject to the civil service laws of a State government, governmental agency or political subdivision. With respect to employment in a foreign country, such term includes an individual who is a citizen of the United States.
The federal law prohibits refusing to hire any individual (i.e. they don't have to be an employee), so you can't use the argument "a barber is not an employee". However, in case some law is stated in terms of "employees", a barber is not generally an employee under the law.
The literal words of the law state that if a person happens to employ 15 or more people in a business that affects commerce (whatever that might be), then because of that aspect of their life, they can never discriminate in the selecting a barber, etc. That is, the law does say "an employer cannot discriminate in that very business whereby they are legally deemed to be an employer". However, so far the federal government far has not gone after an employer hiring 15+ people for discrimination in barber selection.
State laws are also relevant and may have lower thresholds. The Washington threshold is 8.