Applications actually have to render a PDF, and then they send their own rendering to the printer. They aren't just sending "the file" and letting the printer handle it. (That's true for basically any document and application.) One would think that if they look the same on-screen they should look pretty much the same in print, but that's not necessarily always the case. Acrobat also has a lot of options and controls for optimizing output, so even the default settings may change things. (The margins for example seem to be a little smaller on the Acrobat version, which means Firefox was shrinking the page a little more to fit.)
I just tested a print of my bank statement from the two. Firefox made a lot of the text "heavier", not quite bold but a little darker and thicker. But non-text like borders, shaded boxes and logos look the same. On the other hand, when I printed my insulin savings card sheet, everything including logos and shaded areas came out darker with Firefox.
As
@LordDaMan pointed out, different content may be rendered using fonts or bitmaps which may affect it. I'd bet a barcoded label like that is just a bitmap, and Firefox just doesn't do well resizing it. You might make it look better if you adjust the print options like "fit to page" and the margins, or use the system print dialog instead of Firefox's maybe. The label file may be generated for a particular label size, which is why even the Acrobat version looks a little bad because it's having to resize it. See how it looks printing to a normal sheet of paper.
