The way some diamonds (say a 2 carat round VSS1) refract light is pretty damn spectacular, no other rock or material can match it, so that may be a reason why some people want them.
There are a few other minerals that have both higher refractive index and higher dispersion than diamonds, so they could be used to make more spectacular gems than the diamonds.
However the cheaper of them (e.g. rutile, i.e. titanium dioxide) are not suitable for rings, because they have lower hardness so they would be scratched by dust.
Among those hard enough, moissanite could be used for more spectacular optical effects than diamond, though normally it is used in such a way as to mimic diamond, instead of trying to use its superior optical characteristics, to make it stand out.
However the cheaper of them (e.g. rutile, i.e. titanium dioxide) are not suitable for rings, because they have lower hardness so they would be scratched by dust.
Among those hard enough, moissanite could be used for more spectacular optical effects than diamond, though normally it is used in such a way as to mimic diamond, instead of trying to use its superior optical characteristics, to make it stand out.