Answer: it’s two different people. Persian were a people living around the region of actual Iran (Ecbatana). Herodotus reports that these people conquered Asia Minor due the cruelty of the former rulers (mesopotamian, babylonian, assyrian) and to the son of the king’s daughter, who was saved in spite of the fact that after a dream announcing his fall because of the child, the king commanded his death. The child made an alliance with Persian and grouping troops defated his grand father.
Herodotus reports further, that after Greek attempting to introduce themselves in Persian territory (they used to live at the coast), kings of Persia swore eternal hatred to those and started two wars against the Greek (called also median wars), which were both won by the Greek, and subject of the ‘Historias’ of Herodotus, his main work. In the Bible, Persian appear linked to the story of the jewish wife of King (altered names, probably one) Darius, who saved jews from total distruction. (800 to 200 before Christ)
Persian are definitely defeated by Macedonian King Alexandros and the Persian Kingdom is after dissolved.
Macedonian Kingdom doesn’t last very long either and after some quarrels with the arising Roman Empire, it is Constantinos who establishes the Empire in Constantinople. This will give birth to the Byzantine Empire, which actually covers more or less the older Persian Empire.
Around 800 after Christ, a wild amount of tribes grouped under the name of Turk, of the same family than Mongols and Tartars, do arrive from Siberia to Asia Minor. They conquer Iran and Turkey, parts of Syria and arrive even to Viena. Attempts of Byzance to stop their progress, finishes by allowing an establishment of those in Turkey except Constantinople. Around 1500 these tribes group themselves in 12, that correspond to the sons of one of their ‘kings’. One of them is the one of Ottoman, the smallest son of the king. His territory is situated next to Constantinople, in the actual regions of Gölpazari (next to Eskisehir), where he will always keep his horse reserves. He gathers his brothers around the idea of conquering Constantinople, which finally succeeds around 1530. He will become the first Sultan of the Turk, and his long lasting Empire (until 1922) will be called ‘ottoman empire’.