Leave vs. let?

[Elanchana]

But the love you never had is etched in stone forever,
Reminding you of what might have been...

I've been sort of confused lately about the difference between let and leave.

For example:
"Could you let the door open?" To me, this means, "Could you allow the door to open?" But to my mom, it means, "Could you have the door remain open?", which I think should be "Could you leave..."

Could someone help me out here?

The love you never had stays with you like a shadow:
So close, but when you reach, there's no one there...
-Erik, The Phantom of the Opera, Tom Alonso

 

[KingKidd]

Leave is referring to exiting the area, or to keep something in the same situation, being it a door in stasis, a human in shock, a guy having a heart attack...

Let is to temporarily causing a situation, like moving a door so someone can get in and then putting it back.