I'd say the following:
Context list: these house next actions, and next actions only! When you find yourself in that context (e.g. at the shops, at home, at your computer, with your partner, in deep thinking mode, etc) you open the list and pick any actions you want to try to get done. So the formula is "when I am in X context, I check the list to see if I can complete any actions which require me to be in X context"
Checklist: These are for helping you to go through procedural steps, whether that be processes required for performing types of actions, options to pick from, or considerations that need triggering (e.g. how to process a financial transaction, ideas that could be picked from when coming to to write a blog, or things that need looking out for when winter is coming). They are similar to contexts in the sense that they're to be "when I'm in X situation" or "when I'm trying to do X process", but they're not action commitments in and of themselves. Rather, they support you in doing actions or in triggering thinking.