Wikipedia gives one instance of a situation in which complete measures are needed, for the purpose of defining measures on product spaces.
I suggest that you look into Rudin's "Real and Complex Analysis". There he makes an argument that a completion of an ordinary measure space into a complete measure space is just as fundamental to real analysis, as the completion of the rationals to the reals is.
Many theorems in measure theory, for instance Fubini or Radon-Nikodym, needs completeness to make full sense. Fubini is explained in the wikipedia example. To make the other aspect clear -- quite a few statements in measure theory uses the notion of "almost everywhere" -- for instance the definition of $L^p$ spaces, or Radon-Nikodym.
But this notion of "almost everywhere"(rather, "almost nowhere") becomes better if the measure space is complete. It would look really odd if you declare that some property holds true almost nowhere because it holds only on some set with measure zero, and you so arrange things that some other property holds on a smaller set, and then you are no longer able to make the assertion! The product measure example above is a specific illustration in which the concerned property is simply "being measurable", and the consequences are particularly notable.
Added(Jan 16): There are problems into applications into Ergodic theory, for instance. This definition of ergodic transformation and an ergodic theory built on it will run into all sorts of problems if the underlying measure space is not complete. This is again because you need a proper notion of "almost everywhere" and "almost nowhere".