The chief difference between a volcano and a mountain is that a volcano forms around a vent that allows magma to reach the surface of the Earth, while a mountain can be formed by a variety of different geological processes. Most mountains form as the result of plate tectonics, although some were formed by magma welling up from the mantle without breaking through the surface.
Most mountains were formed by the movement and opposition of tectonic plates. When two plates collide, the edges may crumple, forming jagged peaks and valleys. In other cases, one plate may slide beneath its neighbor, forcing the other one upward and forming steep mountains. Magma-formed mountains form when pressure from below forces the crust upwards and creates peaks, but the magma fails to reach the surface.
Volcanoes, on the other hand, form from the accretion of lava after repeated volcanic eruptions. When a vent allows magma to well up from the Earth's mantle and reach the surface, it eventually cools around the vent, forming volcanic stone. Multiple eruptions add to this rock accretion, gradually building up a cone centered around the volcanic vent. If the plate moves away from the hot spot that created the volcano, the volcano may become dormant.