I was going to do a post about quasicrystals and this year's chemistry Nobel, but Don Monroe has done such a good job in his Phys Rev Focus piece that there's not much more to say. Read it!
The big conceptual change brought about by the discovery of quasicrystals was not so much the observation of five-fold and icosahedral symmetries via diffraction. That was certainly surprising, since you can't tile a plane with pentagons; it was very hard to understand how you could end up with a periodic arrangement of atoms that could fill space and give diffraction patterns with those symmetries. The real conceptual shift was realizing that it is possible to have nice, sharp diffraction patterns from nonperiodic (rather, quasiperiodic) arrangements of atoms. The usual arguments about diffraction that are taught in undergrad classes emphasize that diffraction (of electrons or x-rays or neutrons) is very strong (giving 'spots') in particular directions because along those directions, the waves scattered by subsequent planes of atoms all interfere constructively. Changing the direction leads to crests and troughs of waves adding with some complicated phase relationship, generally averaging to not much intensity. In particular symmetry directions, though, the waves scattered by successive planes of atoms arrive in phase, as the distances traveled by the various scattered contributions all differ by integer numbers of wavelengths. Without a periodic arrangement of atoms, it was hard to see how this could happen nicely.
It turns out that quasicrystals really do have a hidden sort of symmetry. They are projections onto three dimensions of structures that would be periodic in a higher dimensional space. The periodicity isn't there in the 3d projection (rather, the atoms are arranged "quasiperiodically" in space), but the 3d projection does contain information about the higher dimensional symmetry, and this comes out when diffraction is done in certain directions. The discovery of these materials spurred scientists had to reevaluate their ideas about what crystallinity really means - that's why it's important. For what it's worth, the best description of this that I've seen in a textbook is in Taylor and Heinonen.