Pure Pursuit (PP) uses the curvature of a circular arc that connects the rear axle location to a goal point on the path ahead of the vehicle. Pure Pursuit was originally designed for fighter plane pilots and tracking. Stanley is a nonlinear feedback function of the cross track error measured from the center of the front axle to the nearest path point for which exponential convergence can be calculated. PP draws arcs to keep the pivot point on the line, Stanley tries to keep the steer axle on the line and the rest will follow. Pure pursuit is a lot like the way we drive down the road, we look ahead to a point on the road and steer towards it. Stanley looks across at the yellow line out the side window and determines the best angle to get back in your lane.
Stanley requires a really accurate heading since the position of the steer axle in the world is determined based on the pivot position and the heading. Pure pursuit generally likes the antenna farther ahead closer to the steer axle to determine distance from the guidance line as it tries to generate that arc back to the line.
PP has many more settings, is generally better for slow speed like under 2 kmh. Stanley works quite well for most ag speeds guidance as it was designed for road vehicles.
Both systems uses 2nd order derivatives (non linear steering methods) to bring the vehicle back to the line. As in, as they approach the line it needs to start steering away from the line to prevent crossing back and forth over the line and never getting on course, like an icy road. the opposite is taking way too long to get back on the line.
Homework time: Here is an excellent read, well worth the time and explains in very good detail how steering systems work Steering methods