But neither human help, nor imperial munificence, nor all the modes of placating Heaven, could stifle scandal or dispel the belief that the fire had taken place by order. Therefore, to scotch the rumour, Nero substituted as culprits, and punished with the utmost refinements of cruelty, a class of men, loathed for their vices, whom the crowd styled Christians.
Christus, the founder of the name, had undergone the death penalty in the reign of Tiberius, by sentence of the procurator Pontius Pilatus, and the pernicious superstition was checked for a moment, only to break out once more, not merely in Judaea, the home of the disease, but in the capital itself, where all things horrible or shameful in the world collect and find a vogue.
First, then, the confessed members of the sect were arrested; next, on their disclosures, vast numbers were convicted, not so much on the count of arson as for hatred of the human race. And derision accompanied their end: they were covered with wild beasts' skins and torn to death by dogs; or they were fastened on crosses, and, when daylight failed were burned to serve as lamps by night. Nero had offered his Gardens for the spectacle, and gave an exhibition in his Circus, mixing with the crowd in the habit of a charioteer, or mounted on his car. Hence, in spite of a guilt which had earned the most exemplary punishment, there arose a sentiment of pity, due to the impression that they were being sacrificed not for the welfare of the state but to the ferocity of a single man.
This is a rather difficult question to answer with any certainty, but I will try and lend a hand.
First, I will be honest by stating there is actually no evidence that Nero even knew Christians existed at the time of the great fire of Rome. Christian's numbered in the hundreds to maybe a few thousand people by this time period, and they were spread across dozens of cities all over the Roman Empire. Simply put, they didn't pose much of a nuisance or threat to Rome at this time because, mainly, no one really knew they existed. It's not until after the turn of the first century that their numbers really began to multiply but even then their numbers were small (it would take another 100 years to gain even 5% conversion rates in the Roman Empire).
Now to answer your actual question, Tacitus is following a common Christian Myth that Christians were hoping to perpetuate at this time. Christians wanted to be known as martyrs because it was part of the saga of Christianity. The idea was that Jesus suffered, and so did his followers and by suffering, you too can be part of this story of redemption. It also significantly help in converting local populaces. To explain it further, I'm copying part of a recent post which is relevant here:
A truly excellent book on this topic was by historian Candida Moss, whom wrote The Myth of Persecution: How Early Christians Invented a Story of Martyrdom. Here, Moss (a professor of New Testament Studies at Notre Dam) argues that there was never a wide-spread persecution of Christians as most people believe. Of the over 250 years of Early Christian history, Moss presents evidence that at most there were only 12 years (at most) where people were harshly persecuted for their faith by orders of the Roman Empire (none of them coming in the first century, during the early conversion times). For it's first few decades, Christians probably went relatively unnoticed. Small churches were known to have sprung up throughout the eastern Mediterranean and even at least one in Rome by 70 CE (this is about when and where Mark wrote his gospel), but it is unlikely Christians faced any real threat yet since they weren't converting huge percentages of people in their populations.
Keep in mind, the period that I was talking about is around the same time of the burning of Rome (64 CE).
As Tacitus wrote these histories, he did so knowing only the information that was accessible to him at the time. Tacitus was writing many decades after Rome burned and nearly 100 years since the historical Jesus died, so he really wasn't in a place to have factual knowledge on either subject. When he wrote about Christian's "hatred of the human race" he was doing so based off what he had been told Christianity was all about. Keep in mind that by this time period, there were many dozens of different Christian Sects, all with various religious beliefs on multiple scales (You had some that were already preaching the doctrine of the Trinity, some whom didn't believe Jesus was a deity, and others who thought that Christ was just one of many gods -- for example), so it's difficult to say what exactly Tactitus knew about Christians at this time.
However, it is likely that Tacitus was told that Christians prided themselves on not acclimating to Roman culture, and that they were, in a sense, anti-Rome, and therefor anti the human race. Early Christians wanted their reputation to be heavily relied upon that they were different than the Romans and that's why they were persecuted, since they hadn't committed any real crimes.
I should also state that a growing wave of scholarship has suggested that part of these writers were actually altered by later Christians. This is easy to do since, like the gospels and most ancient texts, we don't have the originals, and all we have left are copies, most of which are hundreds of years removed from the originals. Scribes, especially in the early Medieval period, had a habit of changing or tweaking older sources to validate other sets of beliefs. Bart Ehrman actually discusses this briefly in his book Jesus: Apocalyptic Prophet of the New Millennium.
Tl,Dr: It's impossible to know for certain since it's likely the texts were changed, however Tacitus was working with the popular belief that Christians were persecuted solely for being different than the Romans due to their spiritual beliefs.