Why exactly were bacteria and archaea kingdoms separated from the now unused kingdom of Monera? Aren't they the same? They are both prokaryotes, so what is the difference?
I'm surprised this term shows up much anymore. Though it's a part of history now, the following paper by Woese demonstrated the three domain system long ago.
This, however, is the net result of the sequenced genomes of eukaryotes, archaea and bacteria:
The phylogenetic differences between these organisms shed light that bacteria and archaebacteria were not as closely related as was initially though, hence the abolishment of Monera.
As you note, "Monera" includes both Bacteria and Archaea -- but genetic analysis and molecular comparisons reveal without a shadow of a doubt that Archaea share a more recent common ancestor with Eukaryota than they do with Bacteria. Consequently, "Monera" is a paraphyletic group, not a proper taxonomic category. Thus the term has been abandoned in favor of the three domain model.
Edit You'll find quite a nice explanation of this at the Wikipedia page for Monera. While modern evolutionary biology textbooks tend not to discuss the now outdated term "Monera", each of the following textbooks lays out the relation among the three domains and discusses the reasons that we avoid paraphyletic groups: Futuyma Evolution 2nd Edition, Herron and Freeman Evolutionary Analysis 5th edition, Bergstrom and Dugatkin Evolution 1st Edition, Zimmer and Emlen Evolution 1st edition, Barton et al Evolution 1st Edition. (Presumably other texts and other editions would cover the same ground).