The program a person uses is based firstly on two major factors:
1) Budget available - free or commercial? How much is there to spend?
2) Deadline - if tight schedule stick to what you know, instead of learning new software.
Only then is the style considered. If the style is considered first, it may have to be changed after considering the answers to the two questions above.
As for the style, there are 3 main aspects to consider (focusing on 2D drawn here, and not 3D prerendered animation sequences or photographed stop motion, however a combination of the two can be used in cutout animation as well) is between frame by frame animation, cut-out animation (tweening) or a blend of both.
You can go for cutout animation in Flash as well frame by frame, or Photoshop and After Effects combined for example as both together work good for cutout.
Some folks even go for Flash and After Effects.