How important is a cooldown jog after hard running?

I am not aware of any systematic studies (at least on PubMed), but anecdotally speaking, in my experience, a warm-down jog seems to reduce muscle stiffness immediately after my workouts. If I stop and sit or lounge about, it is very hard for me to get moving again. I must also say that a shake-out jog the following morning makes the evening run less rickety. So, to answer the OP's question, jogging a few miles makes me feel good. As simple as that.