I've noticed that in the later period of the Ottoman Empire, oftentimes the Sultan and the Caliph would be two different people. Why did this come about? I was previously under the impression that the Caliph and the Sultan were always united in one person so that they were essentially interchangeable.
Why were the positions separated? When did this happen? What were the functional differences in the offices, and finally, were there conflicts between the two for supremacy?
I do not know what list you were looking at that has them as separate people. The Ottoman Sultans had claimed the caliphal title since 1362 and the conquest of Edrine and every Sultan also claimed the title of Caliph.
There is one exception, a Ottoman Caliph that was not Sultan, Abdülmecid II. He took on the title in 1922 when the Ottoman sultanate was dissolved and Mehmed VI was forced to step down. In 1924 the Ottoman Caliphate was dissolved by the National Assembly of Turkey, making Abdülmecid II the last Ottoman Caliph, but he is the clear exception due to the collapse of the empire.
One thing to keep in mind is Caliph/Caliphate is a title that is only as real as the people believe it is. There is not a centralized Islamic church to recognize the title, so it's up to the people that claim the title and the people that follow Islam to accept that person as the Caliph. Also Caliph means different things to the different sects of Islam. So there are different people that claim to be Caliph.
Some will claim that the Ottoman's did not become Caliph's until 1517 when Selim I defeated the Mamluks and took prisoner Al-Mutawakkil III and forced him to publically surrender the title of Caliph along with the sword and mantle of Muhammad. But the House of Osman claims the title since 1362.
But, for the purpose of your question the Ottoman Sultan and Caliph are one in the same with the exception of Abdülmecid II.