This question was actually originally submitted to Movie-mistakes.com, and I think we DID bat this one around a few times after Fellowship released. The answer is, there really were two staffs. (Staves?) The first one had a slot for Gandalf's pipe to fit in, if you'll observe. I don't have time to dig up the exact pictures, but you're sure to find different staff pictures by searching the index of images
here .
Gandalf couldn't go back for his other staff, so I assume Elrond did indeed give him the second one. However, I watched FotR once again some time ago, and noted one important fact. Gandalf ALSO had GLAMDRING with him! You can see it in the brief shot of Gandalf dismounting the horse, and patting its sides as he watched Saruman descened the stairs. So HOW did he get that back?