Roman law during the late Republic and most of the Principate made no distinction in punishment between free people of different social rank. By Roman law all Roman citizens were guaranteed the same legal rights, and the only important distinction in court was whether one was a Roman citizen or not. In cases of civil law (i.e. lawsuits) this was the only distinction, as slaves could not sue or be sued. In lawsuits the nature of punishment (which almost invariably consisted of a fine) could vary, depending on what exact crime had been committed--it was determined by the judge, either calculated by him or drawn from tables. This fine did not change according to the social status of a citizen at court, and such a distinction would've been both impossible and contrary to Roman legal ideals. "Plebeian" does not mean "anybody who's not a nobilis." "Plebeian" just means anyone not descended from one of the original senators, a very tiny hereditary club that got tinier over time and which had already lost pretty much all its privileges by the end of the 4th Century--the Conflict of the Orders is considered to have ended in 287 with the passage of the lex Hortensia, but in reality the major issues (plebeian right to run for magistracy, etc.) had all been secured fifty to a hundred years earlier. By Caesar's day only 14 patrician families still had existing lines, out of more than 50 that we know of originally, and the most of the members of the senatorial class were plebeians. Indeed, since legally one consul each year had to be a plebeian at least half of the consular nobiles were also plebeians. Such men of standing as Cicero, Pompey, Crassus, Cassius, Cato, Brutus, Lepidus, Antony, Hortensius, etc. were all plebeian, and it becomes immediately clear that during the late Republic social status was not equivalent to social order, and that the category of patrician and plebeian cannot have been useful in cases of determining punishment. In cases of criminal law the punishment was generally the same for everyone. As criminal law tried capital crimes (murder, treason, etc.) the penalty was almost invariably death, or occasionally infamia (loss of citizen rights). The manner of execution might differ according to what crime was committed and whether the convicted was a citizen or not, but beyond that there was no distinction.

I should mention exile separately, though. Exile was not an actual punishment during the late Republic and generally during the Principate. During the late Republic exile was not a punishment that could actually be sentenced in court, it was a voluntary punishment. A citizen could voluntarily go into exile to escape the death penalty in a criminal case. He could go into exile before criminal proceedings actually began, but generally exile occurred either shortly before sentencing or in the space between sentence and execution (the trinundinum, which could last up to a month or so). So we see, for example, Milo fleeing to Massilia to escape being put to death for the murder of Publius Clodius. By the late Republic anyone who fled a criminal proceeding or the execution of sentence by leaving Italy was considered an exile, but the status of "exile" was only applied after the fact, not before--after the criminal fled Italy an interdictio would be passed denying him the right to fire and water, that is to say the rights and status of a Roman citizen and a free (or even living, since the death penalty was applied to those who returned to Italy) person within Italy. Officially this was the only type of exsilium going way into the Principate, although in point of fact Augustus introduced a couple of new penalties that, while not legally exile, were essentially the same thing. The most common was relegatio, which existed during the late Republic but wasn't really used. Under the emperors relegatio is used far more often, and it consisted of banishment to a particular place (like the island that Julia was sent to). Tiberius introduced a slightly different version of this penalty, the deportatio. There were also a couple of other kinds of de facto exile that weren't really exile per se or were illegal--this includes fleeing proscription or the illegal (as it was a privilegium) lex Clodia de exsilio Ciceronis that exiled Cicero

This is not to say that social status and wealth did not matter as long as citizenship status existed. That's not really true. Obviously in cases of civil law the penalties imposed on the poor would generally be different (if the law allowed it) from the penalties on the rich. In cases of criminal law, though, all penalties were the same, although Roman citizens could not be executed by certain means (crucifixion). During the later part of the Principate, however, we start to see the establishment of the honestiores and humiliores, social groups that did not exist in the late Republic. The concept probably existed in the late Republic, but it was a social idea that members of the senatorial class or the equites, though largely of the same order as the rest of the population, were not really of the same class--only in the later Principate, especially among the Antonines, does the legal distinction between the two start to emerge. There appears to never have been a legal definition of an honestior, and since by the Antonines the social orders had long since stopped being meaningful and the acquisition of magistracies was no longer a good indication of social rank it appears that it was largely left up to the judgement of the court and the emperor. Honestiores were exempt from certain penalties in criminal cases, particularly a replacement of the death penalty with rogatio (the beginnings of this practice can be seen as early as Augustus). Oddly, by the Antonines it seems that crucifixion had been re-introduced as a penalty for some citizens--honestiores were exempt from it, but humiliores might be crucified, which was not allowed as a punishment for any citizen in the earlier Principate or the Republic. Humiliores could also be sentenced to damnatio ad ludas or ad bestias or ad mortem, whereas honestiores could not. But the legality of these proceedings is kind of fuzzy, and this period is not one that I'm especially familiar with--during the late Republic, which is what I know about, criminal penalty was the same for all citizens, no matter their social rank