So is there a difference between Middle East and Near East? The answer is "nes" or "yo" or "kind of" depending on the year of usage.
The term "Middle East" was officially coined by Alfred Thayer Mahan in 1902 though use of the words has been found from writings from 1900 by British General Thomas Gordon and the term probably existed earlier. The Middle East was used to describe Mesopotamia (modern Iraq) to British India (western limit being where Pakistan is). The Near East was the Ottoman realms in Europe (modern Albania, Kosovo, Bulgaria, Greece, Cyprus with Asia Minor, Syria, Israel, and Jordan).
The various easts (Near, Middle, Far) were made from a European point-of-view (being The West) to divide the various foreign, culturally odd realms from Europe itself. This type of thinking is what led the Greeks to differentiate Europe from the rest of Eurasia.
The Balkan Wars and World War I changed the dynamic of Orientalism (thinking about the easts). Eastern Europe and Ataturk's Turkey were no longer viewed as "foreign" but instead were thought of becoming European. The rump remnants of the Near East were grouped in with the rest of the Middle East. Today those those two terms have been mixed together with no real difference in the use of the term.