At its most basic level, US of A is merely another abbreviation of 'United States of America'.
Rarely, however, is it used as such.
More often than not, 'US of A' is used to enhance a stereotypical attribute of the country.
When used positively, it praises such American features as:
Positive usage permeates all age groups, all sexes, and all income brackets but is mostly used by immigrants or upper to middle class American born citizens.
An example of its positive usage might be:
-"Hey Nancy! Where'd you get your new gun?"
-"You like it? It's a beaut ain't it? Made right here in the US of A!"
In this example, the usage of 'US of A' imparts a sense of dignity and superiority that 'US' or 'USA' would not.
When used negatively, it chides such American features as:
Negative usage is mostly by distraught American born citizens and socially conscious foreigners.
An example of its negative usage might be:
-"Hey Bill! I heard your car broke down?"
-"Yea! I've only had it for two weeks now and the engine failed on me. I guess I should have expected it, it was, after all, made right here in the good ole' US of A. Blah!"
-"I told you to buy Japanese!"
In this example, the usage of 'US of A' imparts a sense of vitriolic disappointment that 'US' or 'USA' would not.
The phrase has a general tendency to be used more frequently in the positive sense within the states and more frequently in the negative sense outside of the states, but of course there are exceptions.