Importance between the American Mathematical Monthly and Research Journals
You're probably right that more people would see something published in the Monthly than in a "research" journal. But if you are aiming primarily to optimize the positive impact this paper would have on a CV for grad school (in the U.S.) I think it would be much better to say that it has been published in a good-quality research-oriented journal, since that will attest much more strongly to the quality of the research (as opposed to attesting to general interest).

EDIT: in contrast to some other answers' implications... The Math Monthly is not at all a research journal. It is an expository and (mostly elementary) problem/puzzle-oriented journal. Publication there is viewed (in my experience, and in my own opinion) as a good thing, but no indication at all about "research potential" or such stuff. That's not the fundamental criterion for publication there!

So, if you are wanting to give evidence that you have already made (and thus in the future can make) research contributions... publication in the Monthly gets you close-to-nothing. Sure, "it's nice", but it's not at all about making significant new contributions to live subjects. It's an entirely different thing. It's not just "a somewhat different journal"... it's in an entirely different category.