For example some were called Claudius, Nero, Trajan etc. while others were called Antonius Pius, Marcus Aurelius, Septimius Severus etc. Why is there this inconsistency? What value does saying "Septimius Severus" instead of "Septimius" have?
They are epithets, similar to Pompeius Magnus (the Great) or Scipio Africanus (the African [conqueror]) - two of the more well known examples of pre-imperial Rome. They mean something like:
Epiteths were usually earned by Romans for their actions, some of them good and others not so much. Gaius Caligula wasn't particularly fond of his nickname for instance, neither was Fabius Maximus Verrucosus who earned that nickname for the wart that grew above his lip.