Counterbalancing helps reduce the natural movement in the clubhead that inevitably happens while swinging golf clubs. This movement can't be stopped but can be controlled with counterbalancing. You can do this when getting fitted for your putter.
A club fitter will have you try a putter, and then they can add different weighted sticks inside the top of the club shaft. The purpose is to find an ideal putter weight for you and counterbalance that weight with the weighted sticks in the shaft.
With the right counterbalance, you will find that your speed control gets significantly better. Counterbalancing offers more stability. The more weight you have in the top of your shaft, the more it will bring you to the center balance of the club. This changes the feel of the club. The extra weight offers more control over the clubs.
It also allows the momentum of the clubhead to do different things a lot easier. There are, in fact, many players that counterbalance all of their clubs. Jack Nicklaus used lead tape to wrap on the inside of the shaft. Adding more weight to the top of the shaft allows you to pivot the club better when you swing the club. This counterbalance makes that easier to do.
Mike suggests that the value of counterbalancing comes down to how it feels. It can create a sensation that makes you hit the ball better. Many players counterbalance their putters; some also do their irons and drivers. Mike puts lead tape on the end of his drivers but not on his irons.