What were Jesus and early Christians like?

[TimOneill]

Can somebody explain the difference between "christ" and "messiah" in the Josephus quotes?

Some translators translate his use of the Greek word Χριστός as "Christ" and others as "Messiah". All mean the same thing - "anointed one". Though "Christ" has more misleading Christian implications for modern readers, so I'd say "Messiah" gives a better sense of what Josephus would have meant: "the one ordained by Yahweh awaited by Jews".

Historians believe Josephus would not have used the word "messiah", but he could have used the word "christ". I looked up the definitions and they seem very similar.

Ummm, no - Josephus would have used neither "Christ" nor "Messiah", given that he was writing in Greek, not English. See above: Χριστός means "anointed one", which is what the Aramaic word Josephus used meant.

Some argue that the wording in the Testimonium differs from Josephus' usual writing style and that as a Jew, he would not have used a word like "Messiah".

Josephus is simply telling us that Jesus was "called" this. He tells us lots of people and places were "called" all kinds of things. Why would he not use the word if this was what Jesus was called? And why would a Jew not use the Greek word "Messiah" if that's the word for the Messiah that he would have found used throughout his Greek translation of the Jewish Bible?

Some of the wording in the Testimonium does differ than that used by Josephus elsewhere. That's because some of the Testimonium has been added later. Some other parts of the Testimonium use language that is distinctively Josephan but is not found in early Christian texts. That's because some of the Testimonium is original to Josephus, as the textual evidence in Jerome, Agapius and Michael the Syrian's versions of it indicates. This is why the majority view of Josephan scholars is that the Testimonium we have today is partly original to Josephus and partly some fairly clumsy later additions.

Which means the only historian of the time who had anything much to say about people like Jesus mentions him. Twice.