A 1099-NEC is used to report Non-Employee-Compensation - usually money paid to a contractor.
A 1099-MISC is used to report miscellaneous income that is received not necessarily in the course of trying to earn money.
Here is more information on the 1099-NEC
Here is more information on the 1099-MISC.
Does it make sense for a mental health counselor to get 1099-MISC forms from healthcare/insurance companies (box 6) then because they are paid by clients this way? Is this worse than a 1099-NEC or taxed differently, or it doesn't matter?
It should not make any difference in that instance. All of the income on those forms should be included in business income if that was the way it was earned.