Yes, different areas of number theory can be very, very different. At my university, I'm surrounded by many different number theorists, and half of us can barely understand the other half. I consider myself a rising analytic number theorist and I'll be working close to 4 other analytic number theorists (Jeff Hoffstein's 4 students and me, a hopeful 5th). And between the five of us, we don't come close to understanding a single thing done by the other major corps of number theorists here (Joe Silverman's 4 students).
To skimp on lots of details, we do analytic number theory and they do algebraic number theory. And this is all different from the number theory I looked into during my undergrad at Georgia Tech. I would now call that Additive Combinatorics or Additive Number Theory, but I think of most of the people who research in that area as analytic number theorists (just different than my cup of analytic-number-theory-tea).
And somehow, there are many disparate parts of number theory. I would say that the three areas I mentioned are also pretty big in drawing lots of attention: the Langlands Program is pretty big right now and has a lot to do with automorphic forms. Elliptic Curves are also pretty big. And Terry Tao does a lot with arithmetic and additive number theory, and thus a lot tends to get done.
But I should mention, to not give you the completely wrong idea, that there is some interplay between the different branches of number theory. I like to think that I'm going to do a very algebraic form of analytic number theory, and questions related to elliptic curves might even come up now and again. But number theory is a large field, all the larger because it tends to be classified by content rather than method (as opposed to group theory, analysis, harmonic analysis, ode and pde, etc.).