I can't find any difference. "uje" sounds a bit too formal for me. I don't remember I've ever used it, but my girlfriend has. May be "uje" is a little closer to "arg".
I cannot make any example to demonstrate the difference. Shame is a kind of social fear anyway.
E.G. You may say both:
Ta oli esinemiseks liiga häbelik/uje. = He was too shy/timid to perform.