Obviously greens and greensites are designed a little differently for public courses that may get 35,000 rounds and a private clubs that may only get 15,000 rounds. At public courses it is important to be able to get folks around the course in good order. Those greensites may have bunkers that are easier to escape and greens that are on the large size.
A Country Club also caters to a variety of golfing skills. Depending on what the founders had in mind CCs seem to have courses that are user friendly. Not all, mind you, but most. That does not mean the course is not well designed but caters to players of all ages and skill levels.
In my experience Clubs that are strictly for golf have the most difficult greensites. Many Golf Clubs get only 10,000-15,000 rounds a year and cater to players that are serious golfers who are either good players or players that want the extra challenge. Greens can be smaller with more slope and undulation, bunkers deeper and the use of waste area and other hazard type places used without someone bellyaching.