I see the term "formal postdoc" used a lot. Anyone care to define that? Is any position where you are recognized as a non-licensed clinician, bill as such, and receive formal supervision to the extent necessary to meet licensure eligibility requirements in some jurisdiction a "formal" postdoc? If not, what is the difference? Is there some registry of "formal postdocs"?
I worked in position(s) following earning my Ph.D. where I received formal supervision. I had the same "official", HR sanctioned job title as other licensed professionals (not using any regulated license-required terms, such as "psychologist") in the same position. Jobs weren't advertised as "postdocs" (or advertised at all, for that matter- I got them through networking and training/clinical connections. My job title wasn't "postdoc," though if anyone asked I would always say "I'm doing my postdoc hours." I was paid based on my experience and the work I did. It was less than others in the same position with more experience, but more than double pre-doc rates (in the 45k+ range, back in the early 2000s). When applying for psych licensure (2 different states) I was able to check the "completed a postdoc" box and have my supervisors sign a form (initial licensure) or just get it accepted as a postdoc (second state, applying with 5+ years of licensure).
As to the OP's question- The biggest practical reason to do a postdoc is that it is a requirement for licensure in many states. You don't know where you will want to or have to end up in the future, so best not to have any barriers to licensure. The biggest clinical reason to do so is that you will (assuming good supervision) refine your existing skills and develop new skills or even new specialty areas of practice. This should translate to offering a better service to your clients. Also, it is a time to potentially develop your professional and business skills beyond what was possible in your grad program, practica, or pre-doctoral internship.