The major difference between prokaryotes and eukaryotes is that eukaryotes have at least three (plants have five), while prokaryotes only have one.
This reflects the many different things you can actually do with RNA. It's commonly forgotten because of the so-called "central dogma" of biology (DNA produces RNA produces protein) that there are actually a whole bunch of different kinds of RNA produce other than messenger RNA, such as ribosomal RNA, transfer RNA and miscellaneous others. Eukaryotes evolved specific RNA polymerases to do specific jobs; prokaryotes have one polymerase that does for everything.
Also, remembering that bacterial genomes are smaller, the requirement of fine control of polymerase activity is much less of an issue in prokaryotes, so they tend to require fewer regulatory factors. These are called "sigma factors", and E. coli (for example) has about seven, compared with the many thousands of transcription factors in eukaryotes.
All RNA polymerases are multi-protein complexes, and the number of proteins that are assembled to form the active enzyme is much larger in eukaryotes; the basic catalytic core is made up of 12 subunits. By comparison, bacterial RNAP has 5 subunits.
The difference in complexity is probably partly down to the complexity of regulation, but probably also has to do with differences in the structure of the genome in eukaryotes (which commonly pack up their DNA into long, linear chromosomes) and prokaryotes (which have a comparatively small, circular chromosomes).
When you get right down to it, the catalytic process of unwinding and pulling apart DNA, pairing off your phospho-RNA nucleotides with DNA and bolting them on to the end of your nascent RNA strand is largely defined by the shape of the DNA double helix and RNA nucleotides. The RNA Pol of Archaea is evolutionarily related to all three eukaryotic polymerases, and I assume also to the RNAP of modern bacteria (i.e. RNA polymerase seems only to have evolved once),
In a roundabout way, I'm saying that the question about whether your magic molecular wand would affect both prokaryotic and eukaryotic RNA pol would depend largely on whether it was affecting the catalytic core where RNA Pol interacts with the DNA molecule - in which case it would affect both - or whether it was targetting sites where the subunits bind to each other, preventing the catalytic core from forming - in which case it would be highly specific to only certain forms of RNAP.
As a final note, just a reminder that the diversity between prokaryotes is can often be greater than the diversity between a given prokaryote and a given eukaryote, so if your magic wand worked for all prokaryotes, my money would be that it worked on eukaryotes as well.