Developer Chris Black made this video showing how Adobe Flash and HTML5 perform head-to-head on a simple Web animation (in this case, a bouncing ball). He uses both an iPod Touch and a Google Nexus One—the latter of which is tested both on Flash and HTML5—to test frame rate and smoothness while zooming and panning.
The results are pretty interesting. Many key figures in the tech industry, most notably Steve Jobs, have forsaken Adobe Flash, mostly used for Web video and animations, in favor of the possibility of HTML5 performing those same duties more efficiently, sometime down the road. Google's Android supports Adobe Flash, though reports on its usefulness have been mixed (most say to only use it when needed for video, otherwise load times will be painfully slow during normal use). But most people haven't seen HTML5 in action compared head-to-head with Flash.
This video shows Flash absolutely crushing HTML5 in frames per second, as well as in its ability to continue to play the animation while zooming and panning. HTML5 is consistently several times slower in framerate, and often stops completely during a zoom, not to mention requiring far more battery life than Flash.
The caveats, well laid out by Wired, are many: HTML5 is a very early project, and probably wasn't optimized properly for this test. Black even admits that some things, like "Gradient fonts, drop shadows, basic video and simple transitions are probably better suited for HTML5." But his conclusion is that HTML5 won't necessarily destroy Flash and that some applications are better suited for Flash than HTML5 (and vice versa, of course). Maybe that's why Apple allowed Adobe back into the App Store.