I pretty much agree with princec here, except in my experience, applets execute slower. Start-up times with Flash and Java are miles apart; Flash will boot up in a split second, but the JRE takes a few seconds and sometimes freezes he computer a bit.
As far as rendering speed… applets are only faster depending on the developer, really. The Java2D library is much slower than Flash (like I said before about AffineTransform). Flash devs don’t have to worry about little niptucks (hw accelerated images, etc.) here and there in their code to see a comfortable speed boost.
A little off topic, but when you compare both platforms, there’s one thing about Java and Flash that sets them apart: fullscreen. Flash supports it, but Java supports it better (and faster). Java’s rendering speed in fullscreen can top some C++ apps. Flash hasn’t optimized fullscreen enough to impress me. It doesn’t give you control of the resolution, bit depth, etc…
To conclude my ramble, if there’s one feature in Flash that I would love to see in Java, it’s the superior FPS control. In Flash you can just set the FPS property and the movie will adjust accordingly. Java is a bit more reliable now that we have things like GAGETimer and 1.5’s nanoTime(), but it took a while for that reliabilty to become available 
But I’m going to get off on that. That’s more of comparing platforms than comparing applets. Blah3 is right, you can’t compare a flash applet and a java applet due to their seperate purposes. But you can compare their platforms. Flash already has a standard GUI, it has recently gotten into networking (besides just loading XML files), and other features that don’t fit the common animation. I doubt the Flash platform will ever be as application-friendly as Java, but we’ll have to see.