For example, could you help me "loosen" or "unscrew" this lid/cap? Which one is correct and more common?
Think of it this way…
If the lid takes 3 turns to remove, then only the first 'grab' will loosen it, but after that it still needs unscrewing the other 2.9 turns
To actually ask someone, either version would probably do - they'd be aware that you were asking because you were having difficulty with that first part, not that you couldn't be bothered with the other 2.9 turns ;-)
"Could you open this for me?" would also work just fine, by the same assumption.
Assuming you mean in normal conversation. They mean pretty much the same thing.
If a asked to loosen a cap for someone I would turn the cap till is was almost off, then hand back the bottle.
If asked to unscrew a cap, I would fully remove the cap, handing them back the bottle and the cap in two separate pieces.
That said, they are really quite interchangeable, and a person may do either one if asked either way. It just depends on the person.
If you mean in some kind of technical document, or instructions it would be normal to see something like
"Loosen the cap until product is accessible, but do not remove the cap completely."
"Unscrew the cap completely to access the product. Once finished replace the cap."
Point being that even in technical documents the loosen/unscrew verbs are usually modified with further instructions to clarify the partially on or all the way off part of the definition.
Unscrew indicates the lid/cap and container are of the screw variety.
Loosen doesn't imply that the lid/cap must be of the screw type, it could be held on by friction.
Loosen also sounds less than completely remove. One might loosen a screw in order to make room for something else to fit on it without removing it completely. Unscrew without any other modifiers has more completeness implied.