In the volume about Tiberius Claudius Drusus Caesar, paragraph 25 he says:
He banished from Rome all the Jews, who were continually making disturbances at the instigation of one Chrestus
Here we have Claudius kicking out all the Jews from Rome (in 49 AD), which incidentally agrees with Acts 18:2. Suetonius tells us that the Jews got kicked out from Rome because they kept breaking the peace in regard to a "Chrestus". Historians widely accept that "Chrestus" is a variant or a reasonable spelling error for "Christ"; but it should be noted that "Chrestus" could also have been a Roman name meaning "good" or "useful".
Here we can see the offensivness or outrageous claim that the Christ (Jewish Messiah) has come. This is something that Jews would find laughable. They would sarcastically ask "You mean the evil has been defeated? we are no longer oppressed? The lion and the lamb will sit together?". No wonder the Jews were causing an uproar regarding the Christians claim.
In the next volume of The Twelve Caesars about Nero Claudius Caesar in paragraph 16 we have:
He likewise inflicted punishments on the Christians, a sort of people who held a new and impious superstition.
This pretty much squares up with the Tacitus reference with Nero inflicting torture on the Christians. This also isn't a glowing reference for Christians (calling is an impious superstition), making the reference more than likely true.
What can we say?
Now both these references don't really tell us anything new. Also these two references are more focused on Christians than on Jesus.
1. Those who followed Jesus were in Rome before 49 AD
2. Those followers of Jesus in Rome broke the peace or upset the status quo there
3. Those who followed Jesus were punished for it