In their 2001 paper, Longstaff and Schwartz suggested a method for American option pricing using simulation and regression, and since then this method has rapidly gained importance. However, the idea of using regression and simulation for American option pricing was used at least as early as 1996, by Carriere. In this paper, we provide a thorough comparison of these two methods and relate them to the work of Tsitsiklis and Van Roy. Although the methods are often considered to be similar, this analysis allows us to point out an important but often overlooked difference between them. We further show that, due to this difference, it is possible to provide arguments favoring the method of Longstaff and Schwartz. Finally, we compare the methods in a realistic numerical setting and show that practitioners would do well to choose the method of Longstaff and Schwartz instead of the methods of Carriere or Tsitsiklis and Van Roy for American option pricing.