We had a similar situation sometime back and we tried moving a 32 bit app (Tomcat based J2EE app) to 64 bit machine. We observe consequence as you are facing.
BTW you don't mentioned why you moved to 64 bit OS :).
Let's take your questions one by one
Does Running 32 bit application in an 64 environment take extra CPU / Memory which is creating issue here.
In general yes. But it depends on your processor architecture as well. To understand this, look at the way Windows runs 32 bit application on a 64 bit machine. There is a subsystem within 64 bit OS called WoW64 (Windows 32-bit on Windows 64-bit)
Primarily there are two ways WoW64does it
Emulating a 32 bit instruction. This computationally very expensive and probably is the cause of CPU usage spike. This is generally the case when used with Intel Itanium (IA-64) based processors.
Switching the processor to 32 bit mode from 64 bit mode. Here, whenever a 32 bit application thread needs to be executed, the processor is switched to 32 mode (compatibility mode) and then switch back to 64 bit mode. This is relatively faster than earlier emulation mechanism.
Is there a performance difference between Win 2003 and Win 2008.
This is very subjective. It depends on the kind of hardware you use. Even if you have 64 bit OS and hardware, a lot depends on the kind of hardware, motherboard CPU etc. 64 Bit OS is designed to leverage superior hardware which a 32 bit OS and machine cannot by virtue of limitations like addressing space etc.
Does Windows 2008 because of 64 bit environment requires more CPU compare to 2003.
Yes. as @EJP mentioned above, it has more things to offer. A more capable OS would need superior hardware. It may be the case the your particular application may or may not need any Win 2008 specific features. That's the reason you need to take a call on why you need to move to 64 bit from 32 bit. Here is similar post on the same forum.
BTW I am not sure if you have an option but you can run the same code without rebuilding (if you are not using any platform specific libraries) on 64 bit Tomcat.