Mark Witton, via flickr
My last post detailed the basic archosaurian anatomical characteristics that form the foundation of dinosaur-kind. They have special holes in their skull and mandible, their teeth are socketed, their femur has an extra knob for muscle attachment, and they have hinged, bird-like ankles.
As this whole deal began with a post on pterosaurs, it's high time we get around to distinguishing the "flying dragons" from dinosaurs; this will also help show the unique features that make a dinosaur a dinosaur and separate them from all other organisms.
Dinosaurs and pterosaurs belong to an archosaurian clade called the avemetatarsalia, which means "bird-like ankle." Basically, their ankles hinge so that they move up and down. Like ours. Other archosaurs have different ankle arrangements; for example, the crocodiles' ankle makes the foot twist frontto back. An older term for the dinosaur-pterosaur clade, fallen somewhat out of favor, is the one I originally learned: ornithodires. This means "bird-necks," and while it's a shorter, more accessible, and just more fun to say, it's not as generally accurate as big old eight-syllable avemetatarsalia (a term coined in 1999 by paleontologist Michael Benton). It's commonly accepted (but not unanimously!) that dinosaurs and pterosaurs are very closely related, sharing a common bird-ankled, bird-necked ancestor who branched off from the crocodiles and other archosaurs (in The Dinosaur Heresies, Robert Bakker even went so far as to propose the idea that this common ancestor may have been warm-blooded - but then, the chapter devoted to pterosaurs is part of the section in which he amasses evidence for dinosaur warm-bloodedness).
The most important difference between the early dinosaurs and early pterosaurs was pretty simple: all pterosaurs flew. A host of anatomical differences required for powered flight are sufficient to divide pterosaurs into their own clade. Of course, you can't simply say pterosaurs flew; dinosaurs didn't as the evidence keeps piling up that some small, feathered dinosaurs took to the air, filled niches the pterosaurs did not, and possessed the right adaptations to survive the Cretaceous extinction. What you can say is that all pterosaurs flew; few dinosaurs flew.
The most important difference that sets the dinosaurs apart from their close relatives are their hips. There were two major hip configurations among the dinosauria; one group, the bird-hips (ornithischians) produced the horn-and-frill-headed ceratopsians, the duck-bills, the tank-like ankylosaurs, and the plate-backed stegosaurs. The lizard hips (saurischians) contained all of the largely carnivorous theropods and the long-necked sauropods. Confusingly, it's the lizard-hips who are believed to have given rise to the birds. What both major clans of dinosaurs have in common, and what truly is unique to them, is an opening in the middle of the pelvis called the acetabulum.
Derivative work based on separate diagrams by wikimedia user Frederik. Licensed under Creative Commons Attribution ShareAlike 3.0
The dinosaurian acetabulum, which is where the femurs meet the hips, was completely open, like a window. Pterosaurs do not have these openings; they are solid. Their pelvises are completely different, in fact; they were not capable of true bipedal locomotion. Pterosaurs are absolutely bizarre and deserving of their own fame, quite separate from that of the dinosaurs. We know very little of their origins and paleontologists are hungry for more specimens of their ancestors. As it is now, even the most basal pterosaur is flight-ready. There are some promising archosaurs who seem to be precursors, but we need more bones.
I'll stress this again: as is usually the case when we discuss fine distinctions between related groups, there is still plenty of debate among paleontologists. As I've noted in the two posts on classification, how the evolutionary relationships fit together is an ever-evolving issue itself. When you're dealing with fossilized bones, and especially the fragile, hollow bones of pterosaurs, it's understandable that there is a lot of room for interpretation. When you pay attention to scientific pursuits, you have to be comfortable with uncertainty.
EDIT 9-02-09: Cleaned up for clarity