What are the differences between tearing and being cut?

Size of thecut. Episiotomies are rarely necessary. It is ok to ask that a routine episiotomy not be performed. A spontaneous tear should it occur may be smaller and less painful than an episiotomy.
Not much. Tearing implies a separate of tissues that are being pulled whereas a cut is usually more deliberate.
Control. Tears cannot be controlled whereas an incision can be planned. For example during delivery tears may extend anteriorly to involve the urethra or posteriorly to extend into the rectum whereas a planned episiotomy will control the direction of the incision/wound.