How much of an advantage does an Applied Mathematics PhD applicant have if he/she has a publication under his/her belt?
Of course, this depends on the "prestige" of the journal that he/she published in. I suppose publishing in a top tier journal like Nature or Science would be a major advantage, but would the applicant be one of the first picks because of it?
More pertinently (I suppose seekers of such advice would not have a top tier journal publication), what about those with publication/publications in lesser journals?
The question is not answerable at this level of abstraction, because grad school admission is not decided on the basis of easily described rules. If you're a coauthor on a brilliant and important paper, you may still be rejected if the committee doubts you were a major contributor to the paper. On the other hand, some applicants with no publications at all may be accepted.
As a general rule, nobody on the committee will read the paper itself. That would be both time-consuming and unlikely to be fruitful, since the committee probably doesn't even have an expert in this specific area. They may get a little information from the abstract or your personal statement, but everything else they know about the paper will come from your letters of recommendation.
The letters need to indicate why this paper should help your case for admission. Specifically, they need to explain why the paper is interesting, what you contributed to it if you are not the only author, and why your work was impressive.
This task will be easier if the paper is really good, and that's correlated with being published in a top journal, but this is not really necessary. For example, if an undergraduate makes a major contribution to a solid but not exceptional research paper, then that could mean a lot, even if the paper doesn't get accepted to a leading journal. Of course the paper needs to meet at least some standards - publishing in a junk journal or vanity press doesn't count. However, the most important issue is demonstrating that you can carry out good research, not getting your name in a prestigious venue.