OK. A CPU has several "parts", but is generally divided into the "core", which is the silicon part that does the actual processing, and the "package", which includes the ceramic or organic structure that you touch, plus the pins, heatspreader, etc. A dual dual core CPU has two CPU cores in one package. It's roughly analogous to having two CPUs, without the expense of having to buy a motherboard with two CPU sockets.
Cache is a part of the CPU that is essentially extremely fast memory that the CPU uses to store data and instructions. Because the cache is so fast compared to main system memory, the CPU can retrieve frequently used data from the cache instead of main memory, which in turns speeds up processing. In general the more cache that a CPU has then the better.
In your specific case, it looks like you would be upgrading from a 2.2 GHz 512k cache CPU to a dual core 2.2 GHz 2x512k cache (512 for each core) CPU. Theoretically you would have twice the processing power, assuming that your motherboard supports the dual core CPUs.
Now the downside is that not all applications will take advantage of the second core. An application has to be specifically written in a way that lets it have two independent processes execute simultaneously (known as multithreading) to be able to use the second CPU or else the second CPU/core will sit idle.
The upside is that Windows NT, 2000, XP, and 2003 are all written to support multiple CPUs, so you will get some degree of benefit even if you are running single-threaded applications. For example, background tasks (like virus scanning) can run on the second CPU while you are running your application in the foreground without any slowdown. You could be ripping MP3s and doing something else at the same time without any noticeable slowdown.
On the even brighter side, you will see some benefit now when moving to dual core CPUs. Intel and AMD have already announced that the future is multi-core, and they are working on 4 and 8 core CPUs further down the line. That means that people who write software will be increasingly writing software that is multi-threaded, so in the future you should be able to take even greater advantage of the second core.