I can't speak for the second half of the question, but the difference between the two terms is that Sultan is a generic political term roughly meaning "ruler", i.e. someone who controls an area of land or a group of people, either as a 'governor' or as a sovereign ruler in their own right. It is important to understand that, as with all words, the precise meaning of Sultan has changed over time, which is why I've given a somewhat vague definition.
In contrast, the word Caliph is a claim to religious, moral, and hence political authority as a successor of Muhammad. It implies direct divine approval. Arguments over who could legitimately claim the title of Caliph have led to many of the most major disputes in the history of Islam, including the most major schism of all, between Sunni and Shia.
For a little bit of backstory: In the beginning, the Caliphate was a more strictly-ran entity, with the Caliphs establishing their client rulers over conquered territories. They had strict codes of conduct, of which much was religious in nature (prohibitions against ostentatious dress, etc.). Most of these Governors (Wali) ruled their districts as both civil and military leader, and were paid handsomely to prevent the chance of corruption, in addition to receiving the spoils of war if they were also the military leader for the district. As the Caliphate got more powerful economically, some of these restrictions began to disappear.
By the time of the Umayyads, the Caliphate had continued to expand and incorporate a significant number of non-Arabs, who were often treated like second class citizens but also retained the ability to operate in the manner with which they were accustomed as long as they recognized the political supremacy of the Muslim rulers. The strict adherence of the many, many Wali throughout the Caliphate directly to the Caliph himself was slowly replaced with adherence to provincial Wali that were given far greater autonomy in tax collection and the selection of civil administrators. Eventually, the collective electoral process of Caliph succession was replaced by the Umayyad's desire for dynastic succession, which was essentially kept until the end of the Caliphate's existence.
The 'Abbasids went along much of the same track, only facing a large degree of fragmentation as pieces broke off to essentially self-govern, or prop up rival states themselves. This produced a significant political problem, as the Caliph himself was to be both the spiritual and secular leader of the Muslim community, and obviously the "situation on the ground" no longer supported this idealized political arrangement.
This was eventually made worse with the success of the Buyids, who forced the 'Abbasid Caliph into vassalage after taking Baghdad; and the Ghaznavids, who wrested control of most of the Eastern part of the Caliphate into their own demense. On the request of the 'Abbasid Caliph (and with a bit of difficulty on account of revolts, the Buyids, and a temporary capture of Baghdad by the Fatimids in 1058); the Seljuks captured Baghdad and Tughril married the daughter of the Caliph.
Why this is important is at the root of the conflict I touched on earlier? It deals with the question of how can the position of Caliph, with its earthly and religious duties, be subordinate to another secular Muslim power.
This was a problem. You had these large territorial opponents in the immediate neighborhood; the rival Caliphate ran by the Fatimids, other powers which acknowledged no authority provided by the 'Abbasids in Baghdad (Like the Umayyad remnants ruling in Iberia), and internal dissent that stemmed from both ambitious magnates and religious discussion over all kinds of things.
In comes al-Mawardi. I'm not going to spend a whole lot of time going over his accomplishments and outlook, but he basically spearheaded a jurisprudence movement that legitimized the role of Caliph as the leader of all Muslims while also being able to lend (if you will) secular authority to others. It also worked to de-legitimize the authority of the competitors on religious claims. This particular mindset is what allowed the Seljuks to come in, take over, and rule the country while acknowledging the authority of the 'Abbasids.
The Ghaznavids were the first to be called Sultan in recognition of their independence, but it was the Seljuks who were the first to be recognized and supported by the political machinery in place within the Caliphate and with Caliphal support. Once that arrangement was made, the 'Abbasid Caliph was always de facto a secondary position, who served only as a spiritual head.
Later, other rulers would take the title and its derivations to essentially signify their autonomy, as Sultan can basically be read as "Strength" or "Authority" or "Ruler," as in No higher authority except the religious lead of the Caliph.
tl;dr - The difference between "Sultan" and "Caliph" involves a separation of religious and spiritual authority (With the Sultan title belonging to a politically autonomous Muslim ruler with nominal connections to the Caliph). The big conflict that created this split is the slow transformation of centralized power towards decentralization, and the jurisprudence movements that sought to legitimize the need of the Caliphate to take less of a role in the field of secular administration.
Bonus: al-Mawardi is really a fascinating character, like many early Muslim scientists, scholars, and philosophers. A lot of his seminal work, al-Akham as-Sultaniyyah, is very boring stuff over the minutiae of governing and Caliphal selection. But, it's very interesting the depths to which he strove to make contenders for control of the 'Umma seem like really bad ideas. tl;dr - He threw shade at the Fatimids by advancing the idea that the "best" person need not always be the Caliph (to counter the Fatimid claim that their bloodline made them the "best" by default), and he went directly against the words of Muhammad to suggest that the Caliph must be a Qurayshi to counter the Mawali/non-Arab rulers of the Buyids and Ghaznavids.