What would be the difference between the public and private graduate schools in US? Okay, so back in my country, the difference between private and public colleges is pretty visible. I have worked in biotech labs of both kinds and know that there's a lot of restrictions on using costly instruments in private colleges whereas we get a lot of liberty to get hands-on lab experience in public ones and sometimes they even encourage the person to conduct independent research without any external interference and to let the person fall and learn. In an internship at a private research lab, I had to ask them multiple times (more than 10 times) to let me use a PCR just once (as I needed it in my project thesis) during my stay there for a good two months. They were doubtful about "someone new" manhandling the instrument despite me having been rigorously trained to set-up that machine in the past (You'll be able to comprehend the annoyance if you have a biology background.) This was one incident and I certainly do not wish to rant about the other ones here. This year, I would be applying to US universities for doctoral studies. I have never been to the US, but I have applied to both category. Should I be concerned about such incidents occurring there in private institutions? Also, here the government institutions are highly reputed and have a very stimulating environment much better than any private institutions.

I have found a similar question before here, but I feel that people mainly talk about funding issues here.
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.