What makes a task a two alternative forced choice task?
I am beginning my answer with this question because there is a general misconception about what 2AFC really means. Many people believe that 2AFC refers to any task where subjects are asked to select one of two options (yes/no, old/new, bright/dim). However what defines it instead is that there are two stimuli presented on each trial.
Your definition can be thought of as an example of a interval-2AFC (AKA temporal-2AFC), but your example is not a typical 2AFC paradigm. It is true that both s1 and s2 are presented prior to the decision, however, typically (in my experience) subjects only respond about one of the stimuli (like my examples in the figure above). That said, the important distinction is that in a true 2AFC task, subjects are presented with two stimuli per trial. Whenever subjects are judging a single stimulus per trial it is considered a discrimination task or a detection task (e.g., judging the direction of motion in a random dot kinematogram).
Pros and Cons of Detection and 2AFC Tasks
The most cited reason for using the 2AFC procedure is that it is unbiased. For example, in my gender discrimination task (presented in the leftmost panel of the figure), a participant could be biased to respond male more so than female. But in the 2AFC version of the task, the participant is choosing a spatial location (left/right) or an interval (1st or 2nd), so any bias they have about gender is eliminated from the decision. If bias is reduced or eliminated it has the added benefit of rendering assumptions necessary for signal detection models obsolete (e.g., that underlying distributions are gaussian).
A second reason for using 2AFC is that there is a performance enhancement resulting from the fact that participants are provided with more information on each trial. That is, in one case you are presented with a single face and must decide which distribution it comes from, which is harder than being presented with one face from each distribution and having to judge which face belongs to the target distribution.
As for the cons, the aforementioned pros may not be true at all. For that discussion I point you to this 2008 paper by Yeshurun, Carrasco, & Maloney.
Ultimately the better procedure will depend on what your question is and what assumptions you are or are not willing to make.