While Quests are pretty much the blueprint of your routine, runs are the actual routines in action. Let's say you created a Quest for onboarding a new team member. You can now use this Quest and create a Quest Run from it for every new team member that you're onboarding. And while you can set default assignees and options for each Quest Run on the Quest itself, you can also set and change them individually for each run, like the assignees on a Quest Run.
All current and previous Quest Runs will be accessible from within the Quest. You will also receive notifications for newly completed Quests by others.