To "let loose" is synonymous with "liberate", "set free" or "unleash".
"The dogs were let loose, and they chased the cars driving by."
"Amy was talking to me for hours, she really let [her thoughts] loose!"
To "run loose" means to go out on one's own, without any previous hindrances or limitations.
"The cat isn't inside the house, I think he is running loose."
"It can be hard for some parents to let their children run loose."