PSG for CQC/most UO maps, AWM for a select few UO maps and all BG maps.
The PSG is superior in pretty much every aspect for close/medium range combat. The chest kill and semi-auto (fast shooting) aspects make it devastating against infantry and snipers who aren't too far away. However, once the distance starts getting larger, the AWM starts improving. The bullet is faster, so you need to lead less on your targets, meaning hitting running targets across a BG map is easier. The lack of bullet drop and increased accuracy also makes it better for counter-sniping, because you can normally only see a tiny piece of a snipers head when trying to hit him. The extra accuracy really helps.
So you need to ask yourself, do I find power/speed more important than long range accuracy?
And yes, the AWM does do more damage than the PSG. If you shoot someone in the arm with a PSG and AWM, the AWM will do more damage. It pretty much kills in 2 hits anywhere on the body. However, the PSG still has the power advantage because pretty much half of the torso = dead.
[edit] In my opinion, PSG > AWM any day. I think it's a better gun, not to mention that the AWM costs like, 3 times as much.