At public schools, e.g. the University of California, International students have a higher tuition rate. Most state (i.e. public) schools have a very low tuition rate for in-state residents, and a much higher tier for non-resident students. In places like California, a grad student who is from, say, Michigan, can attain California residency within 1 year of living in California, reducing their tuition costs. However, an international student, no matter how long they have been in the USA, will never achieve resident status for tuition purposes (you'll eventually be a resident for tax purposes but that is a different thing).
At many public schools, international tuition means that you will cost them 2 or 3 times more money to fund than an American student. Public, state-funded US schools often also have a mandate to educate their own residents, so the combination of this mandate and the financial costs means that at most public schools, only about 10% of graduate students are international. So, if a program has 10 spots each year, it means only one or two international students will be accepted. You will have to beat all the other international students to get that one spot.
Personally, I disagree with ckitc and I would say that their "2" outweighs their "1" but only when "2" applies. That is, when applying to a public, state-funded school or a school where international students cost a LOT more than American students, you will be at a significant disadvantage compared to American students. If it's a very good public school (e.g. UC Berkeley) then you will be competing with many other international students who probably also had an American undergraduate degree. Of course, you will still have a slight advantage over the students who are at schools less known to your admissions committee, but this advantage is very small.
However, when applying to private schools, all students are equally expensive. In this case, "2" doesn't apply and you'll be at an advantage over international students at schools less well known to your admissions committee. At my PhD school, where international and domestic students have high tuition rates, the international student fraction is 47%. Almost half of all grad students are international!
Finally, in your original post, you mention that your F-1 status will be valid for 1 year past your degree? This is not true. Although the initial length of your F-1 status and visa may be 5 or 6 years, as soon as you finish degree requirements, your status will become invalid after 60 days, no matter what your original paperwork says. You could apply for OPT (optional practical training) to extend your visa for the purposes of working in the US in a position related to your degree (e.g. internship, research position etc.) However, this all doesn't matter, if you get into a US grad school, you will get a new F-1 (or J-1) student status for graduate school.