Depends where you are. In the US, the 00 and 01 are identical except for the fake carbon fiber trim in the 01. But in Australia, for example, the new-age WRX was introduced there as a MY01. So if you're buying stuff for your car from Australia, buy stuff for the 2000. It should work perfectly, as long as your 2001 is the old body style.
The new age Impreza introduced in the USA in 2002 is a 2001 model everywhere else in the world. So when you consider parts it's possible that a 2001 part maybe for the 'old face' or the 'new age' depending on whether the manufacturer took the US market into account.