Spasm is a brief, automatic jerking movement. A muscle spasm can be quite painful, with the muscle clenching tightly. A spasm of the coronary artery can cause the pain of angina. Spasms in various types of tissue may be caused by stress, medication, and overexercise.

Source:  Medicinenet

Spasticity refers to feelings of stiffness and a wide range of involuntary muscle spasms (sustained muscle contractions or sudden movements). It is one of the more common symptoms of Multiple Sclerosis. Spasticity may be as mild as the feeling of tightness of muscles or may be so severe as to produce painful, uncontrollable spasms of extremities, usually of the legs. Spasticity may also produce feelings of pain or tightness in and around joints, and can cause low back pain. Although spasticity can occur in any limb, it is much more common in the legs.

Source:  National MS (Multiple Sclerosis) Society