Ok. So I'm pretty much repeating what a lot of other people said. But I'll try to explain it.
A few years ago, Intels best processor was the Pentium 4. It had one processor inside of it. Then, Intel put two of those into one package to make a dual core processor. This merely means that there are two cores on one processor.
So there were two Pentium 4's inside the processor. A little less than two years ago, Intel introduced a new processor. It was a much more efficient design, and it came with two cores in one package by itself. Instead of putting two different processors in one package, it was just built like that natively. These new processors, called the Core 2 Duos beat the older Pentium D's (the dual core Pentium 4's) by a lot.
Since the introduction of the Core 2 Duos, there have been a few less powerful processors introduced, that were slightly different and at a lower price, but still based off of the same architecture. So, they were still dual core processors, but not necassarily a Core 2 Duo.
These less expensive processors typlically had a smaller amount of cache, and a lower Front Side Bus standard.
The models that only have 1mb of cache are named "Dual-Core" prrocessors, to seperate them as having less cache. It really means nothing, other than the less cache.
So, with the 65nm products, we have:
The Conroe Core 2 Duo- 4mb of cache and 1066fsb
The Allendale Core 2 Duo- 2mb of cache and 800fsb
The Allendale Dual-Core- 1mb of cahce and 800fsb
The Celeron Conroe-L single core- 512kb of cache and 800fsb
There are also corresponding 45nm parts, but those are not important in this discussion.
So pretty much these are all based off of the same architecture, but are named differently based on their different features. Pentium D dual core processors are not even close in comparison, and are not worth getting at this time.
About your question, with the only difference in those processors being the clock speed and the amount of cache, one will do better than the other depending on what you are doing. I think I would go with the E2200, because I think the higher clock speed would probably be more beneficial. And also it is a lot cheaper.
But it also depends on if you are overclocking or not.
AMD also has it's dual core processors, but I do not know the timeline of when those came into play, as when I started following computers the Penium D's and the Athlon X2's were already out.
Also I should mention that the Celeron I talked about is a single core processor, I think it has one of the cores turned off.