Two different types of Wingers, usually deployed as wide midfielders, Bale in a 4-4-2 / 4-4-1-1 and Robben in a 4-2-3-1 although occasionally plays more of a forward in a 4-3-3, which isn't really the type of "winger" I am talking about here.
The first:
Gareth Bale. Plays for Wales and Tottenham (FC's) team
A traditional winger in that he is left footed and plays on the left side (usually), he is incredibly fast, a good dribbler, very very direct in his approach. He usually stays out wide on the touchline looking to beat his opposing player (the opponents fullback) to the outside, and either hit in a cross that curves away from the keeper or run at a comparatively narrow angle at the goal for a shot.
Second:
Arjen Robben, the inverted winger. Plays for Holland and Bayern Munich
He is left footed but plays on the right side. He is also fast (not quite as much as Bale), a trickier dribbler, and a little more complete player in terms of passing during the build up. He basically has his own special move; he will cut inside to his left foot, run inside of his fullback, and look to shoot at goal. The fact that Robben does this pretty much every time yet is still possibly the best pure winger in the world is due to both his skill in doing it and that while the opponent knows its coming, not knowing when he will make the cut makes it hard to stop. When he doesn't shoot, he looks to send in a cross that curves towards the goal, this can be more dangerous as it only needs a slight redirection to go in, or can go in on its own, but it takes more time to set up. Robben is willing to go all the way to the byline (end of the field) before making his move, plus he does still have the ability to use his right foot to play in a more orthodox cross.
Overall I don't think one style is inherently better than the other, really any winger should be able to play both sides, which these two can, they just specialize in one. As a coach I don't often use wide midfielders often, but as a general rule the further back in the formation a wide player is the more I would prefer them to be orthodox; if I am using a 3-5-2, the wide players in the 5 would almost certainly be on their orthodox side, whereas if I am playing a 3-4-3 or 4-3-3 the front 3 should be changing positions a lot, but will usually start on their inverted side.