Combining the last thread with this one, i think you have a power supply problem. When failing power supplies are put under a load all sorts of weird things can start happening. I once had one that i could hear through the audio ports, the sound it made would change with the load. It didnt last much longer before it completely died.
Dig up the specs on the new card vs the old regarding power usage. I would not think it would be that much different but you never know. If your power supply is on the edge it might not take much to push it over.
Pull your box out and clean it with compressed air. Use some judgement, you don't want to blow components off the motherboard

. Focus on the power supply and try to blow out any dust or pet hair. Clean all the fans and heatsinks, it is amazing how much crud can pack into cooling fins. Any contamination can snowball a thermal problem. And thermal problems can lead to lots of weird things just like power problems.
For the specific Youtube lockup/crash, disable hardware acceleration in your browser (ask google how for your browser of choice). If you are using hardware acceleration, youtube videos can ramp up the memory clock on your graphics cards and this changes the power consumption. Normally you would not notice this, but again with power/thermal/whatever on the edge it doesnt take much to push you over. If things work well without hardware acceleration, then I think it could indicate a power supply issue. As your cards are fanless i wouldnt think the cards themselves are overheating.
When you installed the new card, did you install the new driver package? As a rule, the driver packages are backwards compatible. But the new card may need some new driver features maybe. I dont have much experience with multiple cards so i am not sure about the best route to go concerning this.
Do these cards use Catalyst Control Center or whatever the latest AMD suite is? If so, you might be able to monitor the temp and power usage of the cards by opening the Catalyst software. Any bit of information might help you narrow down the root cause.
Sometimes it can be really tough to narrow this kind of thing down without doing some part-swapping.