Ivory uses samples. Each key of the piano is painstakingly recorded hundreds of times at different dynamic levels and with different pedaling. When you “play” Ivory, you replay the samples.
Pianoteq is modeled from first principles. It is based on the mathematics of keys, dampers, hammers, strings, and soundboards. When you “play” Pianoteq, you solve a differential equation that describes the physics of an ideal piano.
Neither approach is perfect; both lie somewhere on the upslope of the uncanny valley. Ivory’s samples can’t capture the full quality of pedaling and sympathetic resonance. Pianoteq can’t yet capture the dirt and nuance of real hammers on real strings.
Despite my preference for Ivory, it seems to me that Pianoteq has the brighter future. There’s only so much one can do with samples. Someday soon, perhaps, physical modeling will overtake sampling for instruments as complex as the piano. When it does, we will never look back.