The standard, or most common, 'purple' Linckia is actually Tamaria stria, and has tapered arms and generally orangish tube feet. There is a purple morph of Linckia, but it doesn't turn up all that often.
Orange Linckia may be a variety of things, and can also be a true Linckia in some cases. Again, in non Linckia, the arms are more tapering, and it often has a 'fuzzier' appearance. It is perhaps a species of Henricia, but the differences are difficult to discern for most.
Regardless, these guys are all in the same family of stars as Linckia, and generally have the same care requirments.
Leiaster is very similar in respects to Linckia as well, but I don't think I have seen it around yet. The burgandy Linckia is actually Echinaster luzonicus.
Anyway, again, all very similar beasts.