These expressions have been examined by Aho (Handbook of Theoretical Computer Science, Vol. A, Chp. 5) and Campeanu, Salomaa, Yu ("A formal study of practical regular expressions", International Journal of Foundations of Computer Science, 14:1007–1018, 2003), as well as some of follow-up papers.
Aho calls the more powerful expressions "rewbr" (regular expression with backreferences), Campeanu et al. use "extended regular expression" as well as "practical regular expression". As it seems, "extended regular expression" is the term most commonly used in recent literature.
Building on the term "rational expression" from the French school, and considering the fact that those expressions are used in the real world, I myself like "real expression".
Addendum: A chapter in my PhD thesis deals with this class of formal languages (the corresponding paper is due to appear at STACS 2011). While writing that chapter and the paper, I experimented with various terms. Finally, I decided to use extended regular expressions for the model with backreferences, and proper regular expressions for the nice and normal regular expressions. As it is quite annoying to change the terminology in a paper that is already completely (or mostly) written, I think that some might be interested in the experiences that led to my choice:
First, regex and rewbr don't really roll of the tongue, and using them again and again in the course of a whole paper got really tiresome to write and read, in particular when using any of the possible plural forms. PERL-like regular expressions were also quite unwieldy. Of course, I am no native speaker, so YMMV.
Second, as soon as one wants to talk about both models, it is convenient to use terms that are a variation of regular expression, as this allows one to emphasize similarity or differences as needed (e.g., "a regular expression, be it proper or extended"). Furthermore, this allows one to easily emphasize the special case of "extended regular expressions without backreferences", when talking about special cases in the whole class, instead of comparing different models.
Third, I preferred to use a term that is already used in literature over a newly coined term, which left me the choice between extended regular expressions and practical regular expressions. The second choice implied (at least implicitly) that proper regular expressions are somehow impractical, which felt rather weird (especially as Google's RE2 does not use backrefs, and appears to be quite practical).
Of course, this choice is only my "personal local maximum", and depending on ones needs, other choices might be more appropriate.