Stance
A very good question. I can shoot better using a snooker stance but my back hurts like hell after 5 minuteds of playing this way.
Personally, I will venture out and say that the follow through has a much better chance of being straight as this stance restricts where the arm goes.
This is mainly due to a lower table surface height. Most snooker players have to adapt their stance by either bending their back leg which is normally locked, or shifting their body forward and placing more weight on the table so that the back leg can remain locked.
Of course this is irrelevant to most players under 5'8" in height.
It places dominant eye over cue with less neck twisting, and allows you to use standard stance when up against table.
It is actually more likely that the cue will be central under the eyes. Most snooker players do not rely on dominant eye or are unaware of it without seeking extra input into their game. As long as the cue does not lay outside the eyes then the brain will automatically adjust. The human eye(while still having a dominant side) can be easily trained to sight centrally at distances that are further away. Again there are exceptions like Jamie Jones.
Top snooker players have very good spatial abilities to adapt for depth of field in their surroundings and to concentrate on the table surface as a whole. This is partly due to the green cloth, the table size and also the lit surface with usually dark surroundings.
It could be the same reason that some of the world drives on the right side of the road and others on the left. Snooker is popular in England and the rest of the world wants to be different.
Surpised you didn't say right side and wrong side. Most other countries are aware that in the USA right can actually be wrong...lol
The stance is different because of the difference in table sizes. When one foot is behind the other less reach is available. Anyone who plays all the time on a 12 foot table will adopt the open stance.
This can play a part but again it is easier to 'address' the table with a snooker stance due to the height rather than the length.
Snooker tables are taller and you have to reach farther. Also, snooker balls are much lighter and require less power to move around.
As Ghosst mentioned this is closest to the mark.
The stance itself allows the upper body to position itself so that the shoulders and torso have little chance of moving while moving the rear striking arm. The striking arm also has the best chance of falling into a 'perfect swing'. Upper arm perpendicular to the torso and forearm vertical between elbow and wrist. This position creates a piston-like movement and leaves less chance for any deviation while moving the cue from backswing and through the striking of the cueball.
The follow through to strike the chest is not necessary but can make the finish of the stroke pronounced and therefore guarantee a follow through. Assisting in making a positive shot.
A number of coaches urge beginners to use a mirror to see the action of their arm while in the correct stance. It can actually be quite obvious to see the positive arm action as the stance is corrected to a locked back leg and forward facing torso.