In the first sentence, the fact that the man is running is his defining characteristic. In the second sentence you just need to look at him and the fact that he is running is only mentioned so that you know who to look at.
'A running man' is a bit of a strange thing to say - a man who happens to be running, a man characterized by running. Think of situations where we would use commonly the verb as an adjective like this: a crying baby, a barking dog, a dripping tap, a burning house. It's a baby, dog, tap, house, and something about it makes it more noticeable - the way a quiet dog/baby/tap or ordinary house isn't.