From what i have read the difference between the two Claudii is considerable. I can't find where the split happens but the Claudians are the patrician branch and the Clodians the plebian.

Appius Claudius Pulcher may have prefered the plebian spelling but having Appius Cluadius the Decemvir for an ancestor and Appius Claudius Caecus as a father marks him out as a patrician Claudian

The nefarious Publius Clodius Pulcher (c.95BC- 52BC) is remembered with the plebian spelling even though he was a pedigree Claudian. By his time the split between patrician and plebian was becoming more and more political and we all know what kind of politics Clodius practiced