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."