An acoustic amp, with all "effects" off, is essentially a clean PA. There are some that so a little tone shaping to help make the guitar sound better, but really you want a relatively untouched tone curve. Regular electric guitar amps, even the clean channel, have a significant tone shaping and compression on the signal. So yes, you will hear a big difference. Whether you like it is another story.
So if you run the piezo signal into a true acoustic amp, maybe through an acoustic modeling or "enhancing" pedal, you will get a sound closer to that of a miked acoustic guitar, good enough for "in the live mix", but it won't be exact, of course. Heck, even the piezo from an acoustic guitar sounds "not exact" straight into a PA.
The best small, relatively inexpensive acoustic amp I have tried (and bought) is the Fishman Loudbox. It comes it three sizes, I use the smallest one, 60W. The bigger ones have two separate amplifiers ("bi-amplified") for vox and instrument, while the smaller one has a single amplifier (but two channels to control vox and instrument separately). The bigger ones also have more features, like a notch filter and effects loop.
The small one is loud enough for my purposes, but when we gig we use a PA (the Loudbox has an XLR out to the PA, so I use it as a local "monitor"). If I was gigging solo/duo without a PA, singer-songwriter style, I'd get the biggest one, 180W.