In some fields, people make a distinction between "theory" and "philosophy" (e.g., political theory and political philosophy). On what basis is the distinction made and is there any value in it?

The more theoretical and the more general a subject is, the closer it is to being considered "philosophy." I don't think that there is a definite point where theory leaves off and philosophy starts. So political theory and political philosophy blur together, as do theoretical biology and philosophy of biology, and theoretical physics and philosophy of physics. I think that we worry more about the distinction than we should for intellectual reasons, because we worry about departments and disciplinary categories.

Are there any non-academic journals that someone who isn't a professional philosopher with a degree in philosophy or affiliated with a university could publish a rigorous philosophical paper?

If you have a rigorous philosophy paper, why not send it to an academic journal? You do not need to have a PhD in philosophy or a job as a philosopher to submit to a journal. Best to look for a journal that does "masked" ("blinded") review, so that your identity is not known to the referees and they won't be biased by your lack of professional stature.