Each speaker is a set of designs specs. This is why every speaker sounds different. Every 15" speaker is going to sound a little, or very different. Compare an Ampeg 410HLF to a 410he. They are both 410's from the same company, and they sound very different. That is a result of the specific drivers, and specific cab built around them. Now compare those cabs to the SVT115. The HLF will go a LOT deeper. If the speaker size determined how low a speaker goes, then no 10" based cab could go lower than a 15" based cab. Now look at a cab loaded with K140's. That is a 15" speaker that doesnt have tremendous low end, but it does have a ton of mids.
Driver size does not determine tone.
There are some things that DO correlate to driver size. Efficiency generally does, some times power handling, but that is due to larger motors being put on larger speakers, AND larger speakers having larger displacements than smaller ones.