Precison Shooting concentrates on benchrest, F-Class, Palma, and just generally getting a rifle to shoot better. It also contains some custom gun content.
Accurate Rifle is the follow on to Tactical Rifle magazine, aiming for a broader base of readers. It has excellent articles about Africa hunting by Ken Robinett (a PH and Veternarian over there), and the experiment with handgun articles is permanent with a monthly article by Ayoob. There are also articles about hunting in the US, tactical rifle shooting, reloading and so on. Most articles have a thread about rifle and ammo performance.
There is no comparison to anything on the magazine racks, both of these are superior in every way if you are at all interested in precision and accuracy.
The publisher will send you a free copy if you call up and ask.