Protecting the Peace vs. Enforcing the Law
If you don't put any thought into it... there doesn't seem to be much of a difference here. You could well make the mistake of calling it splitting hairs.
The critical difference is the goal. The goal of a peace officer is for his town to run smoothly, without folks complainin' to much about each other. The goal of the law enforcement officer is to make sure that the public does as it's told.
Law enforcement officers are there to see that the letter of the law is followed, and perhaps the spirit too. Peace Officers are there to hang around and break up fights... make sure violence doesn't erupt without good cause.
For example... A peace officer really doen't give a crap if you're wearing a seatbelt or not. You ain't hurtin' anyone, and you ain't stirin' up a ruckus. A law enforcement officer most certainly does mind. If you're drivin' down an empty stretch of I-40 at 4am at 100mph, a peace officer really doesn't care. You ain't botherin' nobody but the deer. But an LE? He'll take you to jail for that.
LE's worry about the size of the tank on the back of your toilet. PO's worry about very little... least of which is what some politician 300 miles away thinks.
40 years ago we had Peace Officers. We traded them for the "security" offered by Law Enforcement. Freedom loving people around the country lament that fact every day.