We should probably be glad we don't replicate like crystals -- we would be a real mess, then! The form a crystal takes depends on the conditions in which it is grown. Most materials have different polytypes. Carbon can be either diamond or graphite, for example. Silicon Carbide has something in the range of 20+ polytypes (3C, 2H, 4H, 6H, 15R, etc.). If DNA replicated like THAT, people would have to live in bubbles!
As for the difference -- put simply, crystals form the way they do simply because that is the way that takes the least energy in the particular growth conditions. It is something like Legos -- sure, you can put the pieces together diagonally, but they don't tend to stay that way. Much easier to line up the pips with the holes --put it together that way and it stays together. To get a PURE monocrystalline structure, you need very stable and controlled conditions. But there is no information stored in the crystal structure (well, unless humans intervene, of course). The structure is inactive. No crystal organs will form, no tissues, etc.
Could a self-replicating device ever be made? Well, the nanotechnology folks think so. Not yet, though. Would it be alive? Hope not -- that would be kind scary . . . .
Ian