Why some theorems important? There are at least several reasons that come to mind:

{\bullet } Some theorems are important because of their intrinsic nature. They may not have applications, but they just are beautiful. Or they have a interesting proof.

{\bullet } Some theorems solve open problems. Of course any such theorem is automatically important, since the field has already decided that the question is interesting.

{\bullet } Some theorems create whole new directions for mathematics and theory. These are sometimes—not always—relatively easy theorems to prove. But they may be very hard theorems to realize they should be proved. Their importance is that they show us that something new is possible.

{\bullet } Some theorems are important because they introduce new proof techniques. Or contain a new lemma that is more useful than the theorem proved.

{\bullet } Some theorems are important because of their “promise.” This is a subjective reason—a theorem may be important because people feel it could be even more important. Here, both the relation to group equations and the constraints-on-interval-graphs view make us feel Klyachko Car Crash Theorem has some hidden possibilities.


And there is also a paper written by Terry Tao on what’s good mathematics.