US stands for "the United States".
US is short for "the United States".
What are the subtle differences between them?
"Stands for" is normally used when you are explaining an acronym or initialism, and may also be used for other forms of abbreviation:
USA stands for "United States of America"
Laser stands for "Light Amplification by Stimulated Emission of Radiation"
CENTPACCOM stands for "Central Pacific Command".
"Is short for" is more normally used for non-initialism, non-acronym abbreviations, and is also the standard way to explain a nickname:
NORAD is short for "North American Aerospace Defense Command"
Peg is short for "Margaret"
Max is short for "Maximillian"
"Short for" implies that its shorter. "Stands for" just means that A stands in place for B as a verbal alias.
Because these uses go beyond acronyms, sometimes in conversation you will hear things like
"Cookie"? That's short for "the team cook".
"The silver chicken"? That stands for "colonel".
or even, sarcastically (especially in the military, government or other large organizations with their own doublespeak):
"Subdaylight, gravity assisted, vertical insertion"? That's short for "night jump".
In this sense "stands for" is directly replaceable with "means" or "is a/the", whereas "short for" tends to indicate a convenience name which is either actually shorter or is easier to pronounce. In common usage, the difference between the two is often negligible to non-existant.
These are equivalent expressions, but the concept is easily understood (at least in the U.S.) and more usually appears as, "the United States (hereafter, U.S.)..."