Both the Note and the One X will be noticeably snappier than than the DHD. I saw a real difference when moving from the SE Arc to the Note. I have had a decent play on a mates One X and was very impressed by it, but it was not noticeably that much quicker. I think the Note is the fastest of the dual core phones. The thing that clutches it for me with the Note is the size and quality of the screen. Try them in a shop for yourself.
More cores means better multitasking and responsiveness, Quad core vs Single core
More powerful processor, snappier and more responsive (50% faster), 1.5 GHz vs 1.0 GHz
More than 30% more RAM, typically more responsive and better for multitasking, 1024 MB vs 768 MB
londonite
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http://geekaphone.com/compare-phones#
Here's one I did earlier....
http://geekaphone.com/compare/HTC-One-X-Quad-Core-vs-HTC-Desire-HD
More cores means better multitasking and responsiveness, Quad core vs Single core
More powerful processor, snappier and more responsive (50% faster), 1.5 GHz vs 1.0 GHz
More than 30% more RAM, typically more responsive and better for multitasking, 1024 MB vs 768 MB