Wow--I'm surprised by this topic. I thought this was kind of obvious: whether or not you tick the "need financial aid" box has a huge effect on whether or not you're admitted, particularly if you're an international student. The NY Times' education choice blog, and The Chronicle of Higher Education, have covered this substantially. Look up the subject in those two places, and you'll come back with a lot of articles. Anyone who tells you that your ability to pay full freight in this economic climate has no effect on your application is either a) extremely naive, and has been living in a cave for the past 5 years, or b) an admissions officer being unusually cagey.
There are exceptions to this trend, but it is a trend. Your application would have to be better than not most, but absolutely all the other applicants, for you to receive a large amount of financial aid as an international student. And even then.
Some personal anecdotes, which you can take with as much salt as you like:
1. When I went to visit the schools--Tisch and Columbia--the international students I spoke to said they were all paying full freight, with the exception of one at Tisch, who received a $10,000 scholarship. They seemed pretty jaded, one said that they were just there to subsidize the American students.
2. Two of my friends applied to grad schools in both Canada and the US (3 or 4 in Canada, 2 in the US). They did not need financial aid, and were able to provide proof that they could pay the full freight for the American schools (that part of the online form where you give the gist of your financial plan). Both friends were rejected from every school they applied to in Canada, and accepted by both of the programs they applied to in the States. The programs they applied to in the US weren't "easier" to get into--they were their "reach" choices. Canadian schools don't have harder admissions standards. It's just that they could pay--they admit it themselves.
The fact that you're unaware of the impact of the recession--which caused university endowments to deflate rapidly--on financial aid and admission decisions is a bit worrying. There has been a TON of coverage on all the education blogs I regularly visit. Am I the only one who reads these things, as a grad school applicant? More state schools are accepting out of state students to cover tuition, grad schools are accepting international students who can pay full tuition, and anyone who requires financial aid is now at a bigger disadvantage than, say, in 2006.
So yeah, in short: it is a "huge problem". It's existed in academia for years (legacy admissions, anyone?), and since 2008 has exploded. Pay up!