Generally, 'running' is more used in context of tap water/getting water from a mechanical source (tap water, etc.) while flowing specifically refers to the movement of liquid in a free environment.
Leave the water running=Do not shut off the faucet.
Leave the water flowing (Let the water remain flowing)=Allow the water to move out (but cannot be in context of a faucet/sink). You might use this in context of a dam. But you will never say "let the water run" in terms of releasing it.
Flow is very onamonapiatic as well and generally implies an imperfect aspect, as in that something which flows is always having water move out of it, most often in the context of a river but any body of water which moves in a single contained direction.