In the most base level, applications and operating systems are written for a certain bit bandwidth. If an application is written for a 32-bit CPU, it will run on a 32-bit CPU. This has limitations. 32-bit operating systems can only address up to 4GB of RAM. It can only see up to 4GB, anything else is wasted. A single program can only use up to 2GB of RAM.

64-bit CPU's and operating systems can address much, much more. If you have a 64-bit CPU and OS, a program can use more RAM.

Most CPU's today are 64-bit, and 64-bit versions of OS's are common. But in order to take advantage of it, the program itself must be written to take advantage of it. By letting people use more than 2GB of RAM for the game, it increases performance, and stops the crashes alot of people have once the game runs out of RAM, and cannot access more than 2GB.

Okay, I'll take a crack at this one, put that computer science degree to some practical use.

This is going to be a public library referenced example, so if you've never used an actual book library, well I at least hope you understand how it works enough to follow this.

Think of the CPU as being you wanting to read a book. The faster your CPU, the faster you can read.
Think of the RAM as being all the books in the library that you can just get up and go get whenever you want.
Think of the hard drive as being all of the books in the library system (in the city), that you can get any of them you want, but you have to wait for them to send it over to your library.
Think of the 32/64 bit stuff as being like an index that tells you where in the library the book you want to read is.

Lets say that index is on actual index cards, and that each index card can hold the info on where 10 books are in the library. Lets say you have 100 of these cards, that means the index can tell you where to find 1,000 different books. Long as your library is small (not much RAM), that index can easily tell you where everything is. But now lets say your library has 5,000 books, and the index still only tells you where to find 1,000. You've got a lot of extra space there, but since your index can't hold that many books, its basically wasted space.

By increasing the size of your index, you increase how much space you can utilize in your library. Which means if you suddenly need a book on the mating habits of frilled lizards, its more likely to be in your library where you can walk over and grab it instead of having to have it shipped over from across town.

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Another less "explain your computer to your grandmother" approach:

For your computer to access information in the RAM, the processor has to know where in the RAM it is. The more bits it can read, the bigger the "slot number" your computer can look in to pull stuff out of your RAM. If your system can only reference up to three digits in slots, then you can only see slots 1-999, anything in slot 1,000+ is wasted because the processor can't read a number that big to know how to go get it.

By making it so the processor can read bigger numbers (increase the number of bits it can read), it can then access greater amounts of RAM. Which means more of the game code can be stored in the RAM without having to load it off the slow-ass hard drive.

How would performance decrease if you have 4 GB of RAM, but 32-bit WoW only uses 2 GB? Even if it can't do as well as computers with over 4 GB, shouldn't it do better with more than 2?

The game can't use more than 2GB, period. If you are running with 4GB, you have some left over for your OS. So performance won't go down. Unless you use the 64-bit version (AND have a 64-bit CPU, which you most likely do, and a 64-bit OS, which you may or may not) 2GB is all the game can use.

It WILL go down if you only have 4GB and try to use the 64-bit version because you won't have the extra to make up the slightly slower processing of the 64-bit version. If you have 4GB of RAM, stay with the 32-bit version unless you are crashing.

@Drakkenfyre Sorry for posting this late.

But you forget the wow client is now Large Address Aware. Meaning it should be able to access up to 3 GB of memory (3.5 GB on the mac client)

http://us.battle.net/wow/en/forum/topic/1658706617?page=8#145