As the radar typically used conical scanning in order to keep itself pointed at the target (radar lock-on), and the missile guided itself by centering itself in this beam, this caused it to always fly directly at the target.
The birdies sit on a universal joint and thus will point in the direction of the relative wind.
On the other hand, tail-chases can be better for heat-seeking air-to-air missiles, since the hottest part of the target is generally the jet exhausts.
Another disadvantage is that this guidance law tends to require more maneuvering from the missile than laws such as proportional guidance, and that maneuvering may exceed the missile's performance envelope.
Since modern computers allow the implementation of more complex guidance laws with relatively little cost in weight or other design factors, pursuit guidance is rarely used in modern missiles.