I beg to differ about the similar connotations.
In UK slang, being "off your face" means being really drunk or high on drugs. ("Off your cake" meant being E'd up in the early 1990s)
However, planting your face does not necessarily have anything to do with being intoxicated. It just means you fell over spectacularly.
Urban Dictionary says...
Quote:A serious fall in which you land on your face. It's called a "face plant" because it appears as if you're trying to do an impression of a plant with your face as its roots.
Involving sports such as skateboarding, or bike riding, or any other sport for that matter, and you fall, and land directly on your face.
That was a nice face plant when your bike hit that mud spot.