Section 503A is an exemption, not an approval, and it is conditional on four things. A patient-specific prescription; a licensed pharmacist or physician doing the compounding in a licensed facility; bulk substances that either have a USP monograph, appear on the FDA's 503A bulks list, or are components of an approved drug, each with a certificate of analysis from a registered supplier; and the preparation must not be essentially a copy of a commercially available drug. That last condition is the one that moves: it turns on the shortage list, and what was lawful under 503A while a product was in shortage stops being lawful when the shortage is resolved. None of the four requires the finished preparation to be tested, which is the gap that independent assay fills.
Answer first: the two categories differ in whether a prescription for a named patient is required and in which quality standards apply, and that difference decides everything downstream.
Patient-specific compounding requires a prescription for a named individual, is regulated primarily at state level, and is not subject to current good manufacturing practice requirements.
The practical consequences: an outsourcing facility can supply office stock and distribute at volume; a patient-specific compounder cannot do either without stepping outside its category.
Check the registration list. It takes a minute and it is the one verifiable claim here.