The material that transfers energy between the stick and the cue-ball is called chalk. Calcium Carbonate crystals have a certain hardness {and some of the die colors (and manufacturing processes) modify this to a small extent}.
A cue tip impacting a (stationary) cue ball will compress to a certain area {based on the hardness of the tip, the weight of the cue-stick, the weight of the cue-ball, and the speed of impact}. It is this area, covered with chalk, that grips the cue ball and imparts spin durring the millisecond of contact. The chalk grips the tip (leather) and simultaneously grips the surface of the cue ball {Putting microscopic pits on the surface of the ball}. This area is overwhich the power and spin are transmitted.
Thus, the tip you will want is a function of how hard you (want to, have to) hit the cue ball, how much spin you (want to) impart to the cue ball, and what kind of chalk you (want to, have to) use. This is a 3-way (minimum) optimization problem.
Soft tips absorb power while transfering energy from the stick to the CB. Hard tips transfer more energy. Soft tips mushroom regularly and need rather constant maintanence, and wear out rapidly. Hard tips hardly mushroom, last a long time, but you have to chalk with religious furvor. The ideal for many players is a tip that hard enough to allow enough energy to be transmitted, soft enough to allow control of spin transfer, needs little maintanence, and is rather forgiving with respect to chalking. For many players this will be a tip that measures in the 76-84 range on the durrometer.
For a more wide open game such as 9-ball, you might want a (slightly) harder tip to gain power. For a game more like 14.1 continuous, you might want a (slightly) softer tip that gives you the touch and delicate control this game demands.
The kind of leather {cow, pig, water-buffalo}, how it is tanned, how the rest of the tip is made, and how hard it ends up--all vary the feel and the 'hit' of that tip--even if that tip measures the same (as another) on the durrometer.