Slate's Scott Solomon asked some people, and presents a nice, short explanation of why original fossils are important to paleoanthropology:
You can't rely on bone-length measurements taken from a cast, either, since the replicas tend to deteriorate or deform faster than original fossil material. Casts made from plaster or plastic -- such as most casts of Lucy -- often become misshapen in unpredictable ways, especially when they are repeatedly handled or moved. The problem becomes worse when new casts are made from old casts, since a copy of a copy is less likely to be accurate.
And then there's this:
The [Piltdown] forgery was not exposed until 1953, in part because many paleontologists were allowed to carefully examine only casts but not the original "fossils," which contained clues about the subterfuge.
The same goes for reconstructing fragmentary specimens: it is often not possible to tell which details of a cast may be determined entirely by reconstruction, and which are genuine features of the preserved anatomy.
UPDATE(2007/09/19): Wow, a lot of people didn't like this post! Don't read so much into it; the linked article is about why you don't want to lose original fossils. Casts do become misshapen in unpredictable ways. Later methods of analysis can reveal new facts about them. There are lost originals we'd like to have back; like those from Mladeč and Zhoukoudian, all lost or destroyed during World War II. I guess an even better example is Vindija 33.19, which shows that some fossils can be very unique in their preservation in ways that later become very, very important. I've posted some new thoughts related to the correspondence I've gotten.