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.
I can't speak for other academics, but I think it is unlikely to make a large difference which journal you choose. Whenever I assess a prospective graduate student for a PhD program, if they have published papers, I read these to get an idea of how well they can write, and the level of insight in their existing work. (Applicants for graduate school will usually have no more than a handful of publications, if any, so it is not onerous to read them all.) Since I am reading and assessing the content of the papers directly, it really makes no difference to me what journal they are in. Even if the student had published a good paper in a predatory journal with no real peer review process, if the content was good, it would be a good sign for their application. (Although it would mean that the student should be advised to seek publication in a better journal.)
If you are in the US and intend to go to grad school in the US, I wouldn't be too concerned about this. Few undergraduates in the US have any publications at all and so having some record of published results would make you stand out. One advantage of the MAA is that it has wide readership, whereas a more specialized journal will only be seen by a few.
But it is the quality of the work itself, not just the journal in which it is published that is important.
Also, if, as you say, the problem itself is approachable others similar to yourself would benefit from a wide audience.
However, if you aren't close to graduation, then you could explore publication in a more specialized top journal and see what reception the paper gets. You can always send it to the AMM or even JAMS later.
Actually, however, the fact that you are in a position to even ask this question suggests that your acceptance into a good graduate program is pretty much assured. A lot of us got there from more humble beginnings.
Caveat: The EU situation is different, as the European educational system is quite different at the BA/BS level.