As far as the game play on the disks, there is no difference between the two. The Player's Choice version is the exact same game, but a later release. It really was a marketing label used by Nintendo to promote video games that have sold well in the market (over 1 million sold). And much like the first release of a book is a first print, the first release of a gamecube game is actually more valuable. Gamecube games can not be released as Player's Choice, they become player's choice by popularity. The Player's Choice games are usually listed for sale at a lower price and have the yellow labeling on the top portion of the case. Also, resale "value" is typically greater for the original black label versions, just as it would be for an original printing of a book. But as far as the actual disk, normally you wouldn't be able to tell the difference without getting into the code for a date reference.
Just a side note, there are actually 3 different versions of some gamecube games. They are the Players Choice, the original Black Label versions, and a Black Label version with a "Best Seller" gold label printed on it.