Compensating for Change in Thumb Size
Unusually low temperatures in the alley can cause your thumb to shrink, as can playing multiple rounds of bowling. In such cases, inserting some layers of the smooth tape on the part of the hole that touches the back of your thumb can improve the stability of your thumb without providing too much unwanted friction. When your hand heats up and starts to cause your thumb to get bigger, remove the tape.
Better Grip
Textured white tape is used for bowlers whose thumbs are shifting too much within the ball's hole. The textured tape provides greater friction and grip when holding the ball. They usually will put the tape on the part of the hole where the inside of their thumb holds onto the ball.
Smoother Release
Bowlers who tend to have dry, sticky thumbs will often use smooth tape to improve their release. They might insert the tape where the pad of their thumb meets the hole, or along the back side to reduce potential inconsistencies in the way the ball separates from their thumb.
Avoiding Skin Irritation
When the friction of numerous bowling rounds wears at away at the skin of bowlers' thumbs, they will use the smooth bowling tape to reduce this unwanted friction. By placing some smooth tape on the part of the hole that meets the back of the thumb, they can avoid irritating or further hurting their skin.