it's not too different in this situation.
but actually the both meaning is different.
พิเศษ means special, extra, not normal
special friend = เพื่อนที่พิเศษ
เพิ่ม means to add, to increase
add friend (face book) = เพิ่มเพื่อน
sorry if i can't explain you as crystal clear but i hope you get it.