The genus Coprinus was broken up primarily based on DNA evidence in 2001. Fortunately there are also some good macroscopic features that can be used to distinguish the resulting groups. Based on the new evidence if Coprinellus micaceus and Coprinus comatus (the type for Coprinus) were to remain in the same genus you would have to move all of Agaricus and several other genera into Coprinus (or more likely all the Coprinus species would get renamed Agaricus). The four ‘new’ genera (several of them were actually published long ago) are Coprinus, Coprinellus, Coprinopsis and Parasola. Coprinus is very small and essentially only includes Coprinus comatus and a couple of very close relatives. Coprinellus is a large group with a universal veil that breaks up into small particles like Coprinellus micaceus. Coprinopsis includes the familar Coprinopsis atramentaria (aka the Inky Cap). Here the universal veil is more stringy. Finally, in Parasola there is no universal veil. The most common representative of this genus is Parasola plicatilis.

These distinctions are not made in Arora since it was published in the late 80’s. One place you can see the new names is the
Index Fungorum. There is also discussion of these changes on the Coprinus and Psathyrellaceae pages in Wikipedia. Tom Volk also talks about all this in his May 2004 Fungus of the Month page.

Although it doesn’t use the new names, a useful site for identifying members of Coprinus sensu lato is All about Inkcaps

While this research has greatly reduced the number of species in Coprinus sensu stricto, there are actually a number of species in addition to C. comatus still in Coprinus sensu stricto. MykoWeb has a page on one: Coprinus calyptratus (= Coprinus asterophoroides). This page also mentions C. xerophilus.

Based on the the Pacific Northwest Key Council key for Coprinus sensu lato, there are several others in Coprinus ss. (formerly Coprinus Section Coprinus):

C. arachnoideus
C. alnivorus
C. spadiceisporus
C. umbrinus
C. roseistipitatus
C. colosseus
C. palmeranus

The key provides reasonable ways to differentiate between these. The text on the site also mentions Coprinus sterquilinus which is not in the key, but was one of the species included in the DNA anaylses which came out with C. comatus. I have no idea if the other species mentioned above have actually sequenced or even checked for the macroscopic features of Coprinus ss. to ensure their membership.