Please explain when you might use each of these Grade Items. What is the difference between them? I am using 1.9.
Add ... is the label of a button for opening a new grade item form and New ... is the title of the form which the button opens. They refer to manual grade item. See docs http://docs.moodle.org/22/en/Grade_items#Manual_grade_items.
Manual grades items may be useful when you need a column in the grade report which does not have a corresponding activity in the course. Such columns may hold a specialized grade calculation (search down this forum 'formula' to read about such uses) or some external grades which should participate in the course grade (although for such cases you could also add an offline assigment activity which would add the designated item to the gradebook).
hth