"I slipped it underneath the door" implies it stayed under the door. A door is a thing that is not very wide, and "underneath" would be inappropriate.
"I slipped it under the floor" and "I slipped it underneath the floor" would be the same thing, although I would want to use - incorrectly - "to underneath", indicating that "underneath" is a concrete place, whereas "under" is a position. However, I live among non-native speakers who use that construction, so I probably see a non-obvious logic.
"Under the sea" tells me it's in the water, whereas "underneath the sea" is a place below the water.
As for julio's comment, although it's true you wouldn't really say "it's not over it, but under" (realistically) - you can say "it's not over it, but underneath it."