The MKIII is similar to a MKII, but with a loaded chamber indicator, a magazine safety, and a mag release button placed behind the trigger guard.
The MKII is preferred by many shooters, including myself, due to it's simplicity and durability. The MKIII is having a decent track record though. It had a few teething pains, but most new designs do.
For precise target shooting, the Ruger MKII Government Model is/was the way to go. It had faster rifling, and was test fired at the factory for accuracy.
Among the MKIII line, the
Competition Model has replaced the Government. The
MKIII Hunter is also superbly accurate. Both are drilled and tapped for optics if you like.
Another option is a
PacLite receiver/barrel set-up.
To be honest, most Ruger rimfire pistols are capable of out shooting the person who holds them. The pistols with the longer sight radius, adjustable rear sights with a fiber optic front sight or red dot definitely help the shooter harness that accuracy.
You can learn more about them, the MKII in particular,
here.