First things first, I have never owned either car sadly. However, my father in law had the 996 until 3 days ago when it was swapped out for the 997.
Handling: Both cars have excellent manners through the corners. Neither one seems to mind being tossed around a bit. Traction is where the two seem to differ a bit. The new AWD system of the 997 is much smarter than the 996's viscous disc setup. Having said that, I've been slung around in the 996 enough to know that car is glued to the road. We're talking about shades of grey, but better is better.
Now, on to the fun part: the engines:
996 - The power comes on as a growing wave of fury. As the rpms fly past 3400 or so the tubos really kick in and you can feel yourself being pressed harder and harder into the seats. It's almost like the feeling you get when laying down on one of those memory foam beds, only it's not gravity pulling you in it's force fed boxer! It's such a wonderful feeling because the engine seems to grow stronger and stronger as the rpms climb. I love this car!
997 - The 996 is fast. Very fast. But, the 997 is insane. While the 996 delivers it's power in a growing wave, the 997 hits you with it all at once. The sense of lag is almsot all gone thanks to the new variable vane system. I think the dyno shows that maximum torque is reached at 2000 rpms and I believe it! Think of like this: the 996 has lag up to 3400 or so, the 997 has lag up to 2000 or less. It's like getting hit in the stomach with a 5lb bag of flour every time you mash the gas and I like it

. I also need to comment that the 997 is a tiptronic s, and the 996 was a 6spd. The tiptronic setup allows the computer to maintain boost between shifts. The result? Insane power that never goes away

!
My opinion: I know the 997 is faster and better in many ways but I have to say that I prefer the 996's power delivery better. There is just something about the steady increase in power that just feels more enjoyable to me. Both are outstanding cars and you couldn't go wrong with either.
If there are any questions about either I'll try to answer them if I can.