/u/SheepExplosion is right to say that the names simply fell out of use (particularly after romantic naming conventions changed in late antiquity) but it's also true that if we're asking about the patrician families, those had mostly died out by the Principate. The patrician clans of the Principate were artificially created by Augustus, as the patrician order had died out. By Caesar's time only 14 patrician families still had existing lines, out of over 50 families that we know of in the early Republic. From the end of the Conflict of the Orders on the patrician order had rapidly dwindled, and the losses of the civil war accelerated the process. So already by the Principal many of these names were falling out of use or, like Julius and Aurelius, had become more of an indication of status than any blood relation to a noble clan.
As a side note, of the two families you've listed only the gens Julia was patrician--the gens Tarquinia, which died off almost immediately in the early Republic, was plebeian. The Tarquins were not descended from Romulus' original senators, nor were they one of the "Trojan Families" (such as the gens Julia) or one of the Sabine or Latin noble families incorporated into the order in the early Republic (such as the gens Claudia). The patrician order is not the same thing as the senatorial class, nor did they make up even a majority (or, by the late Republic, even a significant part) of the senatorial class during any period following the end of the Conflict of the Orders (and arguably before, as there is evidence that the tradition that before the 4th Century opened up all senatorial magistracies to plebeians only patricians could hold the consulship is probably erroneous--the first consul, Brutus, was a plebeian)
Like most names, they died out when people stopped liking them and using them, e.g. when the family ceased to exist. Some cases, they're just no longer in a recognizable form thanks to linguistic mutations. In others, they fell out of fashion and were replaced by those that were more popular. In Frankish Gaul, for example, we can watch a "de-romanizing" start to occur as we find more and more Germanic names in the record. And from what we can tell, people with "Germanic" names may have been ethnically Frankish or Roman (i.e. Romans adopted Frankish names), but those with "Roman" names were always Romans (i.e. Franks did not adopt Roman names).
This isn't really a field English-language scholarship cares about a lot, but the Germans love it! Sadly, this means the default reference works are in German. In this particular case, the work to read is Ernst Förstemann's cleverly-named Altdeutsches Namenbuch.
Hi! You may be interested in this related post from the other day
Our professor in college explained it really simply like this: It all has to do with the power of the plebeians rising and the power of the patricians diminishing and so they began intermingling and their surnames simply died out and/or were changed through the times. That's also why you don't have people named Priums, Secundus, Septimus, and names like that because plebeian names started to take over. Also Christianity and their names became popular after 313. A.D. so there's that.
EDIT: Fun fact: Patrician families had rules about naming their children so that's why you have so many people with the same name.