What would be the difference between the public and private graduate schools in US?
For graduate school, there are generally few differences. A couple potential factors:


Funding and other resources may be easier to obtain at private schools. On the other hand, there are rich public schools and less well-off private schools. The other factor is that public schools are often larger, so they sometimes have better shared resources (e.g. computing clusters).
Probably the biggest difference is the undergraduate make-up, which only affects you indirectly. At larger public schools you are more likely to TA as a graduate student, which is both good (easy to get TA funding) and bad (may be harder to avoid TAing). 
Some public schools have unionized graduate students/teaching assistants. Almost no private schools have these unions. Whether this is good or bad is an open question.
In very rare circumstances, public school funding can be affected by political winds. There was one recent high-profile case where a large public university's budget was severely cut over a couple of years, but I can't think of any other recent cases. This is extremely unlikely, but possibly a factor if future cuts are being discussed.


The type (public/private) of the graduate program you go to is unlikely to make a major difference in your experience. What matters much more is the quality and characteristics of the program, which is largely independent of public vs. private. Each department is unique, and it's worth closely researching both the research being done and the culture that exists there when choosing where to apply to and where to go.