What is so special about quasicrystals?
It's hard to know exactly what “level” you want address, as the Wikipedia entry seems a decent introduction to me. But in any case, here's what I consider “special” about quasicrystals:

From the fundamental point of view:


Though they are regular (they are formed by symmetry operations that allow them to repeat throughout space), they are not periodic (the symmetry operations are not purely translations). Although there's nothing inherently “wrong” or “impossible” about that from theoretical physics and chemistry, it goes against the classical classification of solids: crystalline or amorphous.
Though hundreds of quasicrystals have been synthesized, there is only naturally occurring quasicrystal phase (icosahedrite)!
All quasicrystals are metal alloys (most of them being binary or ternary intermetallic alloys): given that there are other types of crystalline solids, why aren't there non-metallic quasicrystals?


Regarding applications:


Quasicrystals have found, so far, few practical and industrial applications. In these applications, they are used as other alloys would, and do not have any “magic quasicrystal property”.
One specific feature they have, though, is that quasicrystals tend to be rather strong alloys and that they age well, due to the way they are synthesized