What is this causing the problem?

Most likely its your video card. A 5670 is a good card, but its always been a low end card and is getting older. Seems like when certain effects in the game (lighting, smoke) come into play and your video card has to do more work, its not keeping up.

Your processor is getting older as well, but for gaming you'd see more of an upgrade from your video card than your processor.

For gaming, I'd have to recommend replacing pretty much the whole computer, or at least upgrading the CPU and video card. Switch it out for a Core 2 Quad CPU of at leas 2GHz (and then overclock it, or find a Core 2 Quad CPU that already has a higher clock frequency) if you can find one and a Radeon 7750 or 7770 (or better) for your graphics card. That would solve your gaming performance problems, hopefully without being too expensive. If possible, you might also want to upgrade from 4GB of RAM to 8GB, but that's not as important as the other two upgrades that I've listed.

I would like to be succint: there are two major problems as stated above with you current rig: your GPU and CPU. Your CPU will drop frames because any games with a heavy number of people to render will affect FPS - Saint's Row the Third is no exception. You GPU is as stated, a low end GPU when DX11 first came out - so it's not the best to handle your gaming needs. Just make sure you get something cheap like a HD 7750 to be able to bring up your fps. If your're still not comfortable with the HD 7750, then get a HD 6670 which is pretty sweet for it's performance/price ratio. Otherwise, I would also recommend a Core 2 Quad to extra CPU power while still retaining your current CPU socket. Hope this helps!

What do you mean by go with the 6670 if OP is not comfortable with the 7750? The 7750 is substantially faster than and also uses less power than the 6670. Other than price, there is no reason to get the 6670 over the 7750 for this computer.