Josephus references Jesus twice in his Jewish Antiquities. The so-called shorter passage is regarded by the majority of scholars to be wholly genuine. This is the passage:

But the younger Ananus who, as we said, received the high priesthood, was of a bold disposition and exceptionally daring; he followed the party of the Sadducees, who are severe in judgment above all the Jews, as we have already shown. As therefore Ananus was of such a disposition, he thought he had now a good opportunity, as Festus was now dead, and Albinus was still on the road; so he assembled a council of judges, and brought before it the brother of Jesus the so-called Christ, whose name was James, together with some others, and having accused them as lawbreakers, he delivered them over to be stoned.

This passage is generally regarded to be wholly genuine on the grounds that it seems hard to swallow that a Christian forger would just invent an entire story about the deeds of a high priest, Ananus, following a change in Roman governors, just so he could add a brief and passing reference to Jesus. Therefore, it seems overwhelmingly likely that the whole passage is genuine. The brother of Jesus, James, is attested to by Paul as well - and so here it's very clear which Jesus Josephus is talking about. It's a Jesus who was called (by his followers) the Christ, and who had a brother in Jerusalem called James who was reasonably famous due to being leader of a community of religious believers - in other words, the Jesus referred to here is clearly Jesus of Nazareth.

The other reference, the so-called longer passage, focuses on Jesus specifically, and is regarded by the vast majority of scholars as being partially interpolated but still reflecting a genuine Josephan reference to Jesus of Nazareth. This is said passage:

Now there was about this time Jesus, a wise man, if it be lawful to call him a man, for he was a doer of wonderful works, a teacher of such men as receive the truth with pleasure. He drew over to him both many of the Jews, and many of the Gentiles. He was the Christ, and when Pilate, at the suggestion of the principal men among us, had condemned him to the cross, those that loved him at the first did not forsake him; for he appeared to them alive again the third day; as the divine prophets had foretold these and ten thousand other wonderful things concerning him. And the tribe of Christians so named from him are not extinct at this day.

The reason that it's regarded as being partially interpolated is because it's unlikely that a religious Jew and non-Christian like Josephus would refer to Jesus in such glowing terms and also explicitly call him the Christ. However, there are clues that point to this being a genuine passage at its core. The language being used here is very typical of Josephus (his description of Jesus as 'a wise man' is used by Josephus to describe several Old Testament figures, and is not used by Christian authors to describe Jesus). The 'Christ' reference is almost certainly an interpolation, though most scholars are of the opinion that he'd originally written 'so-called Christ' instead - because he did the same in the James passage above, which is from elsewhere in the same work.

Given the details of this 'Jesus' referenced by Josephus, it's very evident that the Jesus in question is in fact Jesus of Nazareth and not a different person with the same name. We know from Josephus that this 'Jesus' was a teacher with a following who considered him to be the Messiah, that he was condemned to death by crucifixion on the orders of Pontius Pilate, that his followers called themselves 'Christians' and still existed in Josephus' day, and that one of their leaders was Jesus' brother called James (who is also mentioned by Paul).

So the opinion of the vast majority of scholars is that Josephus does indeed refer to Jesus.