I used to find differences in piano actions very frustrating and crippling to a performance, but I now realize that it's mainly the quality of a piano that makes my performance better or worse.
At home, I have an upright Baldwin that's about ten years old and is in reasonably good condition. There are four different pianos at my high school I have played on, two being noticeably better than my piano, and two being noticeably worse. I also play on my piano teacher's, which is a restored Grotrian baby grand, that sounds great and also has a pretty nice action. However, my home piano basically sets the standard for how I judge other pianos. This basically means that if you get used to something, it becomes ok. Other people might hate my home piano, because it has an incredibly bright, even harsh sound (I think the pads on the hammers may be worn down significantly) and the pedal is very muddy and doesn't offer all that much control. The higher tones are also quite piercing, and it's difficult to get a resonant sound out of it.
I know that when I go off to college next year, I will become quite spoiled, because there are four very nice, well maintained small grand pianos to practice on, including a Steinway. An upright is sufficient until you get used to something better. One of the very few material goals in my life is to own a very nice concert grand piano (probably a Steinway), but there's a very good chance that will never happen. I'll probably end up having to settle for something less, because I doubt I will end up being wealthy enough to afford such a thing. Then again, if I decide to not buy a car and live in a small house, there could always be a chance. Heh.
Anyway, I digress. I think that if a piano is of lower quality than your regular piano, both in terms of feel and sound, you will perform worse on it. If it is higher in quality, your performance will be better. Some pianos simply can't accomodate for an extremely skilled pianist who requires complete control over the keys and pedal, or can't supply the depth or greatness of sound needed for a certain piece. All you can really do is get a feel for a piano as quickly as possible once you start playing, and then try to compensate for any shortcomings it may have. Similarly, look for its strong points and use them to your advantage.