How much of advantage do US citizens have in American academia?
True, some federal funding of grad students in the US is restricted to US citizens or permanent residents, but, in mathematics, most funding of grad students is as TAs (=Teaching Assistant) supporting lower-division mathematics courses.

The question of admission itself is mostly separate from federal citizenship requirements. Let me explain some things an admission committee wonders about, statistically, at least, about international applications, that would not be  relevant to an applicant already in the US and with no visa issues: 


It turns out that quite a few applicants simply want to get to the U.S., in effect as informal political asylum, while having limited genuine interest in mathematics, and this creates visa and other complications subsequently. I have no objection to fairly open immigration policies, but I do object to adding that noise to applications to our graduate program. 
Given that the bulk of funding is as TAs, problems with understanding colloquial spoken English cannot be taken lightly. This isn't about "cultural bias", but about the practical features of teaching in this environment.
Minimum funding to generate visa papers: The INS (U.S. "Immigration and Naturalization Services", which controls visas of non-US-citizens) mandates that we promise a certain annual stipend for international grad students... ok... except that that number is several thousand dollars above what the university sets as the ceiling for standard stipends. Not hard to imagine the craziness and injustices this leads to.

Edit: Theoretically, the university is not necessarily responsible for guaranteeing the (needlessly high) minimum, but different universities develop policies designed to keep them at some safe distance from "trouble with INS", and sometimes/often such policies create complications not originally intended, and not due directly to INS. (Also, interpretations and policies drift over years, and I'm refering to a situation from a few years ago...)
International students demand resources for their spouses.  We have repeatedly "discovered" that an admitted international student is married, "needs" a visa for a spouse, the spouse needs admission to something, etc.  It is easy to understand the desperation that leads to such situations, but it also corrupts things, and absorbs resources.

Edit: Yes, if things are done right, as noted in comments, it "should not be" the university's responsibility to "take care of" other family members. Nevertheless, for reasons I do not fully understand, it somehow has come to this on many occasions. (The quasi-legal details are mercifully slipping from my mind! :)
Many other college-university cultures have a hugely different emphasis on problem-solving versus memorization, and relation of teacher and student. Thus, although in one sense the U.S. system generates the weakest outcomes in specialty, because of all the other required coursework for "breadth", the balance of demands, and the goals, of upper-division coursework is mostly compatible with what happens at the beginning of grad school. Further, the less structured parts of the rest of grad school have counterparts in some undergrad projects. In contrast, we have found that several other traditions give students not prototype whatsoever for coping with ambiguity, and this can lead to strange crises well into a PhD.

Edit: Again, not to be claiming that all non-USA programs are the same! For this last point, let's say "traditions which emphasize memorization and reciting, of definitions, theorems, proofs, and examples", as opposed to problem-solving as at least a component of activities. It seems that getting into the habit of thinking that there is a "sacred canon" of results, even perhaps including wording, is not a good habit of mind. Similarly, an exaggerated impulse of "following orders" or "compliance", while a habit highly rewarded in some situations, is not so much the desired methodology after the very beginning of grad school. Thus, success in a prior situation may be misleading to a student, when further adaptation and growth is needed.




Don't misunderstand me: I understand how such things come about. Nevertheless, with limited resources, not only money but energy to help people get in sync with their responsibilities as grad students, there is a serious question about how to allocate. Thus, we can capsulize this by saying that grad school applicants from within the US system are more "known quantities", are already fairly in-sync with the system, are cheaper dollar-wise, and are rarely or never under the threat of returning in disgrace to a genuinely dangerous situation in their home country if we stop renewing their TA!

A small further edit: At a university receiving state funds (although this amount is ever-decreasing), the state legislature, and the electorate, take the viewpoint that the university has an obligation to give priority to students from the region, if not literally the state, and to serve the state's and region's goals. This has become less idealistic during economic downturns, unsurprisingly.

Edit: in response to some comments... I certainly do not claim that "all" non-USA undergrad programs are the same, "bad", etc., nor that all in-USA programs are "good". At the same time, I didn't want to single out specific traditions for criticism. And, certainly, applicants in the U.S. can easily "achieve" weaknesses for no particular reason. As I understood the question, it was inquiring about the realities of the situation, advantages that in-USA applicants tend to have over not-in-USA, at least in the minds of admissions committees.

I do not think literal xenophobia or cultural bias plays any role in admissions, in case people wonder. If anything, the generally stronger undergrad prep of people from non-USA programs has created a "bias" against US students on that criterion alone. The practical and prosaic considerations mentioned above diminish post-PhD, so that "nationality" plays no detectable role in post-doc hiring or tenure-track hiring. (For that matter, examination of the faculty at most "research" universities shows a pretty international population!)