Other than the footwork and skating ability, the stick is the most important tool and skill a goaltender should possess. The stick is the key ingredient for making the rest of the body and it's parts to be in a solid and square save position. If the stick is constantly tracking the puck, then our body is in the desired athletic position. On many occasions we see goalies at all levels including the professionals forget the stick and it will lag behind them whenever they are moving laterally. The stick must track the puck and lead the rest of the body in the direction the puck is going. If the stick tracks the puck it makes you faster laterally, more balanced (to recover for the next save), and provides more net coverage.
Here are some examples of the stick lagging behind and not tracking the puck leaving the body of balance, slower, and with less net coverage...
First notice the large hole where the stick should be along the ice protecting more net. But just as importantly, notice that the muscles on the left side are straining to get to the puck. This is caused by the right side going in the opposite direction of the left side. If he brings the stick to the puck, there would not be the tension we see here making him faster, better balanced, and in a better position to recover for the next save.
The same scenario as above with tension in his muscles caused by not bringing the stick to the puck. Also notice the stick lagging behind causing this goalie to be off balance and it's quite possible he fell onto his stomach thus being out of position for a rebound.
This photo tells the whole story. If the stick was tracking the puck it doesn't go under his leg. If the stick had rotated to the puck instead of lagging behind, the balance would be better, he would have better reach, and in a better position to recover for a rebound.
Below is the correct use of the stick and where it should be positioned...
Notice the stick when moving laterally to the catcher side. Notice the stick is in perfect position filling the 5-Hole as much as possible and the stick is not lagging behind him like in the other photos above. Just as importantly, notice the push with the blocker leg (his right leg) rather than a reach with the catcher leg. Also notice there is no tension in his catcher side arm and leg and his balance is much better for a quicker recovery.