The experience that the Bionic will give is a bunch of waiting to finally use the dual core's full capabilities.
LG Optimus 2X & NVIDIA Tegra 2 Review: The First Dual-Core Smartphone - AnandTech :: Your Source for Hardware Analysis and News
This is hardly a scientific comparison, but I think it's pretty clear that the Bionic will have similar performance to the LG Optimus 2X, and I think it's pretty clear that the Thunderbolt will have similar performance to the MyTouch 4G. In both cases, the hardware is similar enough to draw some comparisons.
You can clearly see that pre-2.4, the performance of nVidia's Tegra is a bit ahead on average... but other than a couple of outliers, the performance appears to be no more than 20-30% higher. Now, that sounds impressive, but not 2x the power kind of impressive. That's what you get to wait 4 months for. Depending on the device you have now, this could be a pretty tough wait.
When 2.4 comes out this summer (let's assume pretty soon after the launch of Bionic), you get to wait for it to come out. Experience has shown this wait could be 2 months. So 6 months from now, your 2nd core will become better equipped to handle multiprocessor situations (and apparently a lot of multithreading specific APIs will open up).
Now, the 2nd core will certainly provide some benefit by taking care of background tasks while allowing your primary core to focus on foreground tasks. However, this is not where the majority of the benefit of dual core will show itself. You have to wait for app developers to begin implementing multithreading in their apps. Many app developers won't bother. Some new apps may come out with multithreading. In these apps, though, you can't expect any more than a 50% performance boost. Developers probably won't have apps for the phone that sport multithreading until at least a few months after 2.4 comes out.
So, you're left waiting for the phone, to wait for the next version of Android to come out, to wait for apps to come out to fully support your additional core. It could be 9, 10, or even 11 months from now!
While, if you choose the Thunderbolt, you get the still impressive power of Snapdragon immediately. Maybe you'll be itching to root, so that's a little bit of waiting. Once root happens, if you absolutely must have extra power, you can easily overclock the processor to 1.3GHz (and beyond) and nearly catch up to the power of the Bionic... before it's even released. And then the Bionic will have seemingly no advantage other than its higher resolution screen. It's very unlikely Bionic will have an unlocked boot loader, so the future of overclocking that processor does not look good.
One extra benefit of immediately(-ish *sigh verizon* ... release it already!

) getting the Thunderbolt is that you'll be eligible for an upgrade sooner than if you waited for the Bionic. And it just turns out that the second generation dual cores will come out 1H 2012... right when you're eligible for an upgrade from a 1-year contract. The second generation dual cores will be faster, significantly more energy efficient, and, best of all, will almost definitely sport Ice Cream. Not only that, but the multithreaded apps will be out there and useful.
It just seems to me that if you get a Bionic you will just experience the pains of being an early adopter. Of course, as long as you don't mind the locked bootloader (I do, very much so), the Bionic is going to still be a great phone. I expect it to have better battery life than a stock Thunderbolt and it will still be as fast as or faster than even overclocked Thunderbolts.
I guess I weigh getting a new phone NOW and enjoying it NOW as being more important than dual core. And the last thing I want is to have a locked down phone after I've been spoiled by how awesome a rooted phone is.
(wow, that got a lot bigger than I thought it would) ... inb4that'swhatshesaid. >_>