The benefit of the left to right drag is that performed correctly it should allow you to go past the opposition on their reverse side. While not always this is generally the weaker side for the tackler, which means that it tends to be a more consistent option.
Although not shown in the video below it is also possible to include a 'dummy' before dragging from the left, this can be either by moving the stick over the ball as if moving it left or by some body movement.
This can draw the defender into committing the wrong way. By varying these it can also make it difficult for the defender who will often look for certain cues to read what the attacker is likely to do.
With practice you can vary each of these different skills with different movements to disguise what you are looking to do.