Okay, first off, what are type and creator codes? Are these the method by which OS 9 and earlier open files?
Yes, that is what OS 9 and earlier used to keep track of what type of file something was, and what application was meant to open it by default. This information is part of the resource fork of a Mac file. For example, every file saved from Photoshop gets the Creator code of 8BIM. With that information, the Mac knows any file you double click on the desktop which has that creator code will cause Photoshop to launch (if it's not already running) and open in that application. The Type code signifies to the application what type of file it's supposed to be. An .eps file saved from PS will get the Type code EPSF.
Second, what is UTIs if they are not the same as type and creator codes?
http://developer.apple.com/macosx/uniformtypeidentifiers.html
http://en.wikipedia.org/wiki/UniformTypeIdentifier
Why would you need file extensions if you already have a method to tell the computer what application to open a file with?
They are used in conjunction. If you didn't have the UTI metadata, you'd run into the same problem Windows has. Once the .jpg extension is assigned to an application, then it doesn't matter what program actually created a .jpg, they will all try to open in the same app. The UTI data allows the Mac to know which app created that .jpg so it opens in the correct app, even though you may have multiple apps that can save a JPEG image.
Third, I have snow leopard installed on my macbook and I can still open all of my word and excel files, all of which lack file extensions, so what exactly has changed?
They have Type and Creator codes, which Snow Leopard still understands and will use them if there are no extensions or UTI data. So they still open in the correct application and carry the correct icons assigned to them.
Finally, what exactly is wrong with the old method of opening files anyway?
Wish I could find the article I read explaining why Apple wanted to do away with Type and Creator codes at some point in the future.
OS 9 opens files and applications a whole lot quicker than OS X does on both of my computers...
OS 9 was a very compact system in comparison, so code executes more quickly. It also had a lot of strikes going against it. It was already a hodgepodge of spaghetti code in the attempt to keep it up to date with the myriad of Mac hardware from all the way back to the first Macintosh along with numerous changes to file types and software. The biggest hit against it was that it was pretty much impossible to ever bring that old code to 64 bit. Add in almost no support for OpenType fonts, Unicode, almost no security to speak of, no protected memory ability so any app that crashed brought down the entire system, etc.