Any different symptoms of MS and nerve damage in the back?

Some differences. Normally nerve problems in the lower back (where they are most common) cause pain, numbness, weakness or decreased reflexes in the legs. They are often asymmetrical. The symptoms last a long time. MS often causes numbness or weakness on one side of the body, double vision, or impaired vision in one eye. These symptoms may only last a few days.
Pain may be present. A "pinched nerve" in the back can cause radiating pain down a leg. Some ms pts get diffuse leg pains and cramps. A ruptured disc can cause numbness and weakness in one or both legs, and can seem similar to ms. A neurological exam however, can sort out spinal cord (ms), and nerve roots (low back issues) for example.