Reference/Introduction to partial   difference(NOT differential!)  equations The title says it all. Despite heavy googling I have not been able to find anything. What I am interested in, is theory   (maybe modelling), not for the moment finite difference methods as approximations to partial differential equations! Books, papers, webpages, ....
 A: Partial Difference Equations by Sui Sun Cheng?  A good start (in the context of reading the classics) is the paper by Courant-Friedrichs-Lewy.  Also useful is a list with more books.
For the non-partial case, I must admit I have a soft spot for Milne-Thompson's book.
A: I don't have a complete answer, but the math physics literature has many papers on "discrete Schrödinger operators," which are partial difference operators on a discrete space.  These are analyzed in their own right, not as an approximation of some continuum model.  Look for "discrete Schrödinger operators."  Also look for "tight binding approximation" or "tight binding Schrödinger operators."  Also try spelling Schrödinger as Schroedinger.  
I don't know of a "canonical reference" along the lines of Evans for PDE's, say.  On the other hand, the basic theory of discrete difference operators (existence, uniqueness, etc.) is usually easier since there are no local regularity issues.  
A: There are a couple of books that describe how asymptotic methods developed for differential equations can be extended to difference equations
Bender Orszag Advanced Math methods points out that even the simplest nonlinear difference eqns are rarely exactly solvable and also do not exhibit movable singularities (unlike PDE).  No general procedure exists to determine local behavior of nonlinear difference equation.  There are very few techniques for closed form solns.  Some of the widely applicable techniques are substitution, use of known nonlinear functional relations and use of generating functions. 
Holmes in Perturbation Methods Sec 4.8, 3.9 and 2.7 discusses how WKB, multiple scales and matched asymptotic expansion can be applied to solve difference equations.
See also http://www.math.niu.edu/~rusin/known-math/99/difference_eq for two books which lucubrate on the analogy between difference and differential equations
