Without wading into the sea of half-information that's been presented and correcting/clarifying it, I'll just say that there's no way anyone can generalize that ALL 1x15's and ALL 4x10's are going to sound a certain way. There are many 1x15 cabinets that don't have any meaningful response below low E on a GUITAR and there are 4x10's with a legitimate 6 dB downpoint at low B on a five string, and the best way to determine what will work for you is to hear it for yourself.
Typically, most 4x10's are voiced to emphasize midrange, low mids on cabs like the old Eden XLT series and upper mids on more "hifi" cabs like an SWR Goliath or Trace Elliot 1048. 15" speakers became ubiquitous for bass cabs but they're not always the best choice if you need more low end close to you onstage. They can project large amounts of low end into the audience while you end up sounding thin onstage. That's why cabs voiced like the Eden XLT's became popular. Even though they have little to no true first-octave response, their heavy low midrange makes for a very "bassy" sounding cabinet that can make you sound more full onstage without throwing out a larger amount of PA muddying bass.
On the subject of tweeters, there is a psychoacoustic phenomenon where increased bass leads to the perception of increased high end because the brain "extrapolates" a more complete harmonic series based on the added low end, but the caveat is that this requires actual low frequency content and not just more low mids. This is also why tweeters became standard issue on 4x10's; they were overall more muddy sounding because of the lack of actual low end, so tweeters were added to give more tonal articulation.
I think in drbassman should check out a bunch of cabinets, but I have a feeling that he will be best served by a 4x10.