Seem to be synonymous. However, on closer examination it seems that "rush" has a nuance of haste and carelessness, whereas "hurry" seems only to carry the nuance of speed. 1.I rushed up the stairs. 2.I hurried up the stairs. Speed Up: An increase in speed, especially in a person's or machine's rate of working.

hurry - i brought you curry in a hurry rush - hush and rush! speed up - increase the speed of the activity