I was reading this page: 【所】 と 【処】 はどう違いますか? | HiNative and was a bit confused on XGpmn's answer. The impression I'm getting is that 処 is more of an old-fashioned term, but are there other more major differences?
(I think I know that 所 can be used alone, but 処 is only a word extension/suffix)
処 has multiple meanings in the dictionary: dispose, manage, deal with, sentence, condemn, act, behave, place.
処理 and 対処 are "deal with", yes. But 処分 is much closer to "dispose", 処罰 and 処刑 are from "sentence"/"condemn", while 処女 comes from "place" (originally used to describe a woman living in her home before the marriage).
In short, it differs from word to word. There's a reason all of those are listed... 🙂