Fundamentally, the Patriarch and the Caliphate are different things.

The Greek Patriarch is an ecclesiastical (non-temporal) spiritual leader. The Emperor Justinius first codified the Pentarchy, which was formally recognized in 692 AD. The Pentarchy was an establishment of five Patriarchs (which could also be called Popes), which would be seated in the five most important cities to the religion: Rome, Alexandria, Jerusalem, Constantinople, and Antioch. The Great Schism which would sunder the Catholic faith from the Orthodox faith was due to a large number of factors, including claims of preeminence of the Patriarch of Rome. The Schism would divide the Christian faith into three; the Catholics in the West, the Orthodox in the northeast, and the Coptics to the south. While the Roman-Catholic Pope would establish a temporal state (the Papal States), the Orthodox and Coptic heads of faith did not establish temporal rulership, meaning they were never heads of state as well as heads of faith.

The Caliphate, by contrast, is both a spiritual and temporal title. Muhammad, the founder of Islam, established a faith and an empire, and his successors were chosen in a non-hereditary process with democratic elements. This next bit is going to focus on specifically Sunni points of view, as the Ottomans were Sunni, but please keep in mind that many of the following points are disputed.

The first Caliph, then, to Sunnis, was Abu Bakr, Muhammad's first successor, and the first ruler of the first Caliphate, called the Rashidun Caliphate. The title of Caliph is a claim to be Muhammad's proper successor as temporal ruler of all followers of Islam as well as the proper head of faith of the Muslim faith. Many Muslim rulers have claimed to be the Caliph, but historians generally recognize four Caliphates: Rashidun, Ummayad, Abbasid, and Ottoman.These four empires ruled a sufficiently large proportion of their specific school of Islam as to be effectively temporal and spiritual ruler of that school (the Ummayad not really ruling the Sunni world, as they were based in Iberia rather than Arabia/Egypt). Depending on your specific brand, historians may recognize others, but those four are largely undisputed from a historical sense (although again, from a religious stance, wars have been fought over whether Abu Bakr or Ali was the first true Caliph, and continue to be fought to this day, so this should not be read as absolute fact).

What does this mean?

Abolishing the Caliphate aligned with Ataturk's goal of secularizing Turkey. I won't speak specifically to his long term goals and intentions; while I have an admiration for the man, I haven't studied him extensively, so someone else would have to speak to specifics of his view of religion in society, but I can say with confidence that he did not want the head of faith to be the head of state and vice versa. The Greek Patriarch didn't threaten that goal in the way the Caliphate did.