While Seigan is in effect, the recast time for Third Eye is halved.
Seigan also adds a chance of Third Eye to remain in effect after anticipating an attack, instead of wearing off after just one. The chance of remaining in effect seems to decrease over the course of Third Eye's duration, with the highest chance being available immediately after Third Eye's use.
Seigan also adds the possibility to Counter an attack along with anticipating it.
Third Eye Is not required to Counter attacks, but vastly increases the chances. It is possible to receive a Counter while Seigan is active and Third Eye is not, but it is quite rare compared to how often it happens out of "Anticipated" attacks.
Seigan (正眼) (Lit: Angle of the eyes) is a basic defensive stance. Specifically it is the preparation for an enemy's attack by aiming your sword at their eyes. This stance is also done with the left foot back and the right foot forward, giving the defender a much better ability to step backwards while limiting their ability to attack. It is common to assume this stance immediately after drawing.