The open stance generates the most racquet head speed and requires the least time to get in position, but it demands greater precision to make clean ball contact and control direction, and it's awkward to hit while moving forward. You can hit with blistering speed if you just smack a ball flat with an open stance, but it makes more sense to use a good portion of your high racquet head speed to generate topspin, which will give you some margins for consistency.
The square stance is easiest for beginners, but it's also used at the highest levels of competition. A square stance presents the most linear alignment between racquet and ball, offering the greatest certainty of ball contact and directional control. This alignment makes it easier to hit while moving forward: your swing path parallels your running path. It takes a little more time to prepare a square-stance shot, though. The lower racquet-head speeds of a square stance make flat and slice shots more likely to stay in, but you can also hit strong, although not usually severe, topspin.
Not surprisingly, the semi-open stance offers a nice balance between the open and square stances. You get some of the angular momentum that generates racquet-head speed in the open stance, but you also have some of the benefits of the more forward-moving, linear motion of the square stance.
