It's mostly a stylistic choice based on how each word sounds.

I would use 'to run' for water, and 'to flow' or 'to go' for rivers and alikes. Using 'to run' for a river will emphasize something. Only the Author knows whether there is some meaning to be emphasized.

Have you ever heard this song: The ballad of Easy Rider ? First lines run like this: The river flows It flows to the sea Wherever that river goes That's where I want to be Flow river flow Let your waters wash down Take me from this road To some other town ... ... This is the BEST version: https://www.youtube.com/watch?v=3dvBjKr2LcY it starts at 35 min. 30 sec. Hope you enjoy this song!