Why is the class NP-Complete important compared to NP-hard?
There are at least a few reasons that NPC is interesting:


The class NP contains many problems that are interesting (both practically and theoretically), moreover many of these problems turn out to be NP-hard (and hence NP-complete), but many problems outside of NP are almost certainly too hard to be of more than theoretical interest, so NPC provides a (rough) group of problems that are apparently hard, but not so hard that we can't try to do something with them.
In other words, NPC is probably the limit of what we can hope might be polynomial-time solvable, it would seem a stretch to try for PSPACE = P (for example).
The class is NP is structurally interesting. It's the basic example of "do we get any more computational 'speed' from nondeterminism". So we're interested whether P=NP or not, and NPC is (probably) an important component of working that out.
NP-hard (as a class) is really too big and varied to deal with as a single thing, it's everything that can be reduced to from an NP-complete problem, including a huge swathe of stuff outside NP, so from the point of view of trying to develop general results and techniques, there's nothing to grab on to.