One major advantage of going to a state school over a private school is the amount of money that you save. I have spoken to a few people that go to SUNY Stony Brook and their tuition per year is something like 12,000 and here at NYU the tuition is 48,000/yr. So in a state school, you will save a ton of money.
Now with private schools, you might be exposed to newer technology and materials since these schools don't really depend on state funding as much as state schools do.
As far as specializing, as Tom said, if you have the grades, the skills, the experience, and the NBDE scores good enough for PG programs, you will get in anywhere. The only thing being is that some private schools reserve a spot or 2 for their own graduates.
Hope this helps,