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In short, crystals differ by their chemistry and/or symmetry.
Crystals are solids whose atoms are arranged in a repeating pattern. Table salt, NaCl, is a common example. Ions of sodium and chlorine are arranged in a regular three-dimensional grid. The symmetry of table salt can be generalized as cubic.
Quartz, SiO2, is another common example. Quartz is composed of SiO4 tetrahedra that also follow a repeating pattern. However, instead of cubic, quartz is either in the trigonal or hexagonal system. This is evident in nicely formed quartz crystals, which tend to have six prism faces.
To compare, quartz and salt have chemical differences (SiO2 versus NaCl) and symmetry differences (trigonal/hexagonal versus cubic), making them two different crystals (I prefer to say two different minerals).
Two crystals can be different and still have the same chemistry, though. For instance, olivine and ringwoodite are both (Fe,Mg)2SiO4. But olivine is orthorhombic, and ringwoodite is cubic.
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