I have both also. Its because some use the old 32 bit version, some use 64. You can use both at the same time. Its because some browsers, such as firefox, are only 32 bit. You need to get both.
@Op, try running it via minecraft.net. What browser are you using?
To clear up the confusion: 64-bit versions of Windows are capable of virtualizing a 32-bit environment in which they can run 32-bit programs. Given that the vast majority of programs that are out there are 32-bit, there would be no way that Windows could exist in the consumer space with just 64-bit compatibility. Java comes in two versions, 32- and 64-bit, which allow it to run compatibly with both 32- and 64-bit programs. Any purely java program will be capable of running in either the 32- or 64-bit JRE. The only difference is when you need to have browser plugins (for example), which require 32-bit versions of Java for 32-bit programs (like Firefox) and 64-bit versions of Java for 64-bit programs (like IE 9 x64). There is no conflict to have both installed simultaneously.
@OP: Are you using the .exe or the .jar version of Minecraft to launch the program? Try using the other one. If you are using the .jar version, try it with both the 32- and 64-bit versions of java.
I am having the same problem I believe on a nearly identical machine. I had been running minecraft fine for over a year until four days ago when this stuff started happening. Nothing anyone has suggested has helped me so I installed a fresh copy of Windows and everything ran smoothly until I performed a windows update and teat is when everything went to pot again. I am currently trying a few more things and if I get anything figured out I will be sure to post it here.
Reinstalling Windows is one way to fix it for sure, but it's not ideal. I doubt that it's a Windows update, otherwise many more people would be having problems. Anyways, start minecraft from scratch (backup/delete .minecraft), re-download the exe from the site, and also make sure you have the latest ATI Catalyst drivers. If that doesn't work, then create a new account on windows (through Control Panel) and try playing minecraft on it.