The difference is that between the general and the particular. An errand is any short trip to do a job or task for someone. Shopping is one kind of errand. Other types of errand are possible, such as taking a message, finding something out, collecting something from somewhere, etc.

A possible difference could be that between the general and the particular. An errand is any short trip to do a job or task for someone. Shopping for someone else is one kind of errand. Other types of errand are possible, such as taking a message, finding something out, collecting something from somewhere, etc.