Adjacent angles are two angles that have a common side and a common vertex (corner point), and don't overlap. Vertical angles are the angles opposite each other when two lines cross
Vertical means upright and is perpendicular to the horizontal plane, which in day to day terms means the ground if we can assume for a moment that the ground is flat at least locally around us. It can mean overhead - of or at the vertex.
Adjacent on the other hand means the one next to it, its neighbour, also may mean bordering or close or nearby. It is near but not necessarily in contact although often you may find they are touching but that is perhaps more precisely then described as adjoining.
