This image shows the forces acting on the bob of a simple pendulum:
This image shows the forces acting on the bob of a conical pendulum:
In the first one, $mg \cos \theta = T$, and $mg \sin \theta$ is the restoring force. In the second one, $F \cos \theta = mg$, and $F \sin \theta$ is the restoring force. Why does that difference exist?
In the second scenario, a centripetal force is required, perpendicular to the motion of the bob, which only the horizontal component of the tension in the string can provide. In the first scenario, the restoring force in the simple harmonic motion of the simple pendulum acts towards the mean position, which is provided by the corresponding component of weight. Hence, $mg$ acts as the restoring force. They are two very different scenarios.
