Really really, the difference is negligible at the moment. No graphics card can utilise it properly yet.
This is the kind of difference you will be looking at
v2.x: 500 MB/s (5 GT/s)
v3.0: 1 GB/s (8 GT/s)
So yes technically it has twice the "performance" ( as the description on the motherboard touts) But if no card can use it, whats the point?
I think we will have to wait for the 7XX & 8XXX cards before we see any real gains. All the cards out at the moment take very little from running on pcie3. So i wouldn't worry.
I'm pretty sure there isnt incompatibility issues between the 3820 and the 670. Can you link any of the reviews that said the 3820 was compatible with PCIE3? Becuase i think hookie nailed it on the head. I have seen people claiming to use PCIE3 with a 3820, but i think that might be with a different bios for the mobo. IMO it really is not worth it. You will not notice any performance differences while playing.