I have two instances where speaker size seems to matter. Anecdotal yes, but here they are. First, understand that my pair of 4x10s (Peavey, Ashdown, both with horn) are great for electric bass. They handle my Tobias 6 string just fine. Ok, onward.
1. I play upright bass. I have a Crate BX100 combo amp with a 15” speaker that sounds far better with the mic’ed upright than does my bass stack. I’ve tried using a 4x10 extension cabinet instead of the 15” for this, and it does not sound as good. That amp with that 15” sounds like the bass, only louder.
2. I run a Hammond organ emulation app into a rotary horn (MotionSound Pro3T), which has left and right low rotor simulation outputs. It sounded good thru those 4x10s, but I also own an actual Hammond A100 and Leslie… and something was missing from the sound of the Pro3T low rotor outputs. The Leslie uses a 15” for its low rotor. So I took the Peavey out of the emulation setup, and added a GK Backline 1x15. It fleshed out the low end, and now my Hammond emulation sounds right.
From my standpoint, the 15” clearly moves the air differently. My hypotheses: for the Hammond, nothing emulates a 15” speaker better than a 15” speaker! For the upright, its own sounding surface is very large - to best reproduce that, a large diameter speaker is the best option.