Apologies for just now getting back to responding to this; I've had stuff to do.
Same here. A good PA is somewhat akin to a stereo. What goes in comes out louder. I think it's all in the capture.
An IDEAL PA would do that, but very few PA systems reproduce sound that accurately. Great systems come pretty close but many systems that could be considered 'good' fall considerably short of that ideal.
The reason I think your bass is never going to sound the same through your PA as it does through your bass rig is that the two systems express completely different design goals and concepts. Your bass rig is designed for one purpose, to amplify the bass signal, using drivers selected for that purpose. Your PA speakers are designed to amplify everything, more or less, but with a heavy emphasis on vocal reinforcement. The two-way design, with a compression driver and horn, will yield a very different sonic signature from the start. But in addition to that, all manufacturers of speakers in this category use a great deal of signal processing to coax additional performance out of the relatively small package, as well as to ensure the whole system is relatively 'blowup-proof'. The compression, limiting and EQ algorithms applied in the DSP yield a sound that IS apparently louder but quite a bit different from the input signal. This is not to throw shade on your particular system; it's characteristic of all the speakers in this category, although some implementations are better than others. When things get loud, your bass rig is going to sound fine -- like a bass rig. But your PA rig is going to sound compressed, with apparently reduced headroom and dynamic range. This is addition to the equalization curve that's likely implemented to emphasize the vocal range. Although this could maybe be EQ'ed out, it doesn't do much to enhance a bass signal, while your bass speakers, running essentially full-range, uncompressed, and exhibiting natural response roll-off, sound more like what a bass SHOULD sound like, at least IMO. This strays somewhat into the area of systems being designed for sound production vs. those designed for RE-production -- a system intended to be part of the characteristic sound of an instrument rather than a wide-range sound system intended primarily to amplify vocals and acoustic instruments.
Disclaimer: I was an audio engineer and system designer for my entire working career, but I'm retired now and haven't heard any systems like this that have been designed recently. Maybe they've gotten better, but I'm skeptical. DSP and the close coupling between amps and drivers in a self-powered system enable a lot of performance advantages, but the various implementations can produce audible inaccuracies. Designing any sort of speaker system involves balancing a number of compromises and trade-offs, resulting in different systems that perform quite differently and that may perform one particular task better than another.
