To "let loose" is synonymous with "liberate", "set free" or "unleash".
Some examples:
"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.
Examples:
"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."