In The Great Gatsby, in what way are Nick and Gatsby similar and different in Chapter III?

You have of course highlighted an important aspect of the
novel, which is narrated using the unreliable first person narration of Nick Carraway.
There is a sense in which both Nick and Gatsby are kind of "doubles" throughout the
novel. This chapter is interesting for a number of reasons, but firstly because it
charts the first actual meeting between the two characters. One similarity that it
quickly established is the shared history that both of them have through fighting during
wartime:

We
talked for a moment about some wet, grey little villages in
France.

As for another
similarity, you might want to think how both Gatsby and Nick to differing extents are
part of this party, yet at the same time are not really part of it. For example, Gatsby
is described as "standing alone on the marble steps" during the piece of music "Jazz
History", just as Nick struggles to enjoy the party and make sense of the dazzling world
he has just entered.

As for differences, you might want to
think about how Gatsby "chooses his words with care" and adopts the English phrase "old
sport" to try and fit in to this world, whereas Nick describes himself as an honest man
at the end and does not change himself to fit in. Clearly other differences, all
witnessed through Nick's eyes, are the tremendous wealth he sees that Gatsby enjoys, the
massive sense of speculation concerning Gatsby and his past and finally the sense of
speculation about where Gatsby gets his wealth from and how he has achieved so much in
so little time.