Generally, a 部活動 is supposed to be more serious. If it's a sport club, the members have training sessions in the morning before lectures and also after school, probably at least three times a week, and participate in tournaments, etc. etc. Also, I would expect that it has a coach or manager in charge; it's more official.

If it's a サークル活動, the members may play the same sport as above, but just for fun in principle. サークル (or 同好会） simply refers to a group of people with a same hobby, gathering periodically to enjoy it together.

I'm not sure what English terms would be the most suitable translations for them, but that's how I differentiate them in my native Japanese.

Hope this helps.