The identity of Trachodon remained a mystery until the discovery of good remains of Hadrosaurus in New Jersey, which Leidy described two years later, in 1858. Teeth associated with the Hadrosaurus skeleton were similar to those of Trachodon, and paleontologists entertained the idea that these two genera might be the same. Note that they still didn't have a good skull, so they didn't know about the duckbill; basically they considered Hadrosaurus and Trachodon to be advanced relatives of the European Iguanodon.
Numerous teeth similar to those of Trachodon and Hadrosaurus continued to be discovered in the American and Canadian West for many decades, and these two genera acquired numerous species. Other hadrosaur tooth genera were proposed as well, such as Cionodon and Diclonius. Paleontologists had a field day referring any of the species in any of these genera to any of the other genera. The lack of decent skeletal remains made for much confusion.
Finally the first good skull showing a duckbill and skeleton were unearthed and were described by Cope as Diclonius mirabilis in 1883. He used Leidy's original species name, Trachodon mirabilis, but gave it his own generic name, Diclonius, which he had coined earlier for other hadrosaur teeth from the Judith River area. He considered Trachodon to be an invalid generic name that needed to be replaced. Actually, Cope was correct in considering Trachodon to be doubtful, but his name Diclonius is equally doubtful, and it was certainly unwarranted to suppress Leidy's genus in this fashion. But that's Cope for you.
The nomenclatural problems raised by the overuse of Trachodon and other dubious hadrosaur tooth genera were, at the beginning of the 1900s, so tangled that when it came time to mount Cope's "Diclonius mirabilis" skeleton at the American Museum of Natural History, the staff left it unlabelled, calling it a "trachodont" skeleton. By then the genus Trachodon had been used as the basis for the whole duckbilled-dinosaur family, Trachodontidae; hence the museum label.
It was another few decades until Richard Swann Lull and Nelda Wright published their monograph on North American hadrosaurs in 1942. There they reviewed all the previous species, decided the tooth genera and species were mainly junk, and coined the fresh name Anatosaurus for the commonest North American flat-headed (noncrested) duckbilled dinosaur. The type species of Anatosaurus was Anatosaurus annectens, based on a nice partial skeleton with skull that Cope's rival Marsh had originally described, using his own hadrosaurian genus, as Claosaurus annectens in 1892. Cope's "Diclonius mirabilis" skeleton became the type specimen of a different Lull & Wright species, Anatosaurus copei.
This is how matters stood for many years, until the pace of discovery of new hadrosaur skulls and skeletons made it apparent to Mike Brett-Surman (and unindicted co-conspirator Jack Horner) that Anatosaurus annectens was just too similar to another hadrosaur species, Edmontosaurus regalis, to remain in a separate genus. Since Edmontosaurus had been created by Lawrence Lambe in 1917, it had priority over Lull & Wright's Anatosaurus, and Brett-Surman sank Anatosaurus as a junior synonym in 1979 or thereabouts (actually, first in his unpublished M.Sc. thesis in 1975). He kept the species separate from the type species of Edmontosaurus, as Edmontosaurus annectens (actually, since I was already corresponding with Mike at the time, I'm the first to have published the combination Edmontosaurus annectens, in Mesozoic Meanderings #1, 1978, but I did it with his blessing and with reference to his dissertation).
But Anatosaurus copei was a different story: it was too different from Anatosaurus (=Edmontosaurus) to remain in that genus. So a new genus was coined for that species (also in Brett-Surman's thesis, but not published until 1990, in an appendix to an article on hadrosaur morphometrics by Ralph Chapman and Brett-Surman). The species became Anatotitan copei. The name Anatotitan was originally suggested by Donald Baird, then Jack Horner's mentor, and Mike Brett-Surman used it.
When it came time to revamp the American Museum's dinosaur displays earlier in this decade, the mount of Cope's skeleton (one of two at the museum, incidentally, positioned side by side in the Cretaceous hall) was finally relabelled from "trachodont dinosaur" to Anatotitan copei, which it is my guess it will stay for quite some time.
As for Leidy's original Trachodon teeth, modern analysis suggests that one is hadrosaurian and the other ceratopian; the hadrosaurian tooth is presently considered the type specimen, so Trachodon remains a hadrosaur. But Jack Horner has analyzed a multitude of duckbilled-dinosaur teeth during his career, and he (along with Lull & Wright) thinks the type tooth may belong not to a flat-headed hadrosaur but to a crested one -- a lambeosaurid, possibly Corythosaurus. Horner also sees similarities with Prosaurolophus (a noncrested hadrosaurid) teeth. Unfortunately, there is no way to identify which genus of lambeosaurid this might be, so Trachodon will remain a doubtful lambeosaurid (or maybe hadrosaurid) genus into the foreseeable future.