When you have to "step up" to an unfamiliar piano brand in a recital, or you are just playing it in private, there can be several differences including timbre, responsiveness of action, relative tone decay, pedal adjustments, ability to control dynamics, projective power, etc. But over time, the unfamiliar becomes more familiar.
In my student days (i.e., when I was much younger!), I could bet that whereever I played, it would be either a Steinway or Baldwin, as those were the two standards of quality in the U.S. It didn't take me long too figure out the idiosyncracies of both instruments, to anticipate them, to exploit their respective strengths, and to compensate in any manner necessary. Once in a great while I'd encounter a Mason & Hamlin, but that wasn't too mysterious, as my teacher owned one along with her Steinway. Today, though, the student is apt to face off with a Kawai, Yamaha, or other brand as well, complicating things further.
Having said all this, the other fact of life is that even pianos of the same model within a brand can have surprising variability too. These differences are much harder to anticipate until you actually start playing and figure out the instrument as quickly as possible. This is the very reason that Horowitz took his own piano with him for public performances. He dreaded these uncertainties and risking a less than stellar performance. But Artur Rubinstein and other notable pianists could take the best and worst pianos thrown at them and somehow make lemonade out of lemons whenever necessary. Unlike the flautist or violinist who can conveniently carry their instrument cases around, we pianists face a much more formidable challenge in frequently facing the unknown.
Topic: Performance quality from piano to piano