MAA Monthly vs. AMS Bulletin
The American Mathematical Monthly (which is I think what you call the MAA Monthly; often abbreviated AMM) is an expository journal at a level slightly above undergraduate -- i.e., it is a journal for mathematical papers (and problems!) that most mathematicians can understand (as opposed to specialized papers written for experts in a specific topic).
The Bulletin of the AMS (short BAMS) is a graduate-level and research-level expository journal.
In theory, there can be overlap between the scope of the two: An exposition that is sufficiently elementary to be understood by undergraduates, yet sufficiently novel that it is of interest to researchers, could be published in either journal. (For example, various parts of combinatorics come to mind.) Also, both journals publish papers on the history of mathematics sometimes.
The American Mathematical Society also has another expository journal, called Notices of the AMS, but it does not directly compete with the AMM either; it is more focused on brief (proof-less) surveys for researchers than on anything suited to undergrads. It also focuses on contemporary issues of the mathematics community.