When medieval Christians wanted to end a marriage, they had to petition the Pope for a papal dispensation. A dispensation declared that a previous marriage was void because it either violated canon law (e.g. if the people involved had been too closely related) or did not meet canon law's conditions for a fully realised marriage (i.e. the marriage was never consummated). This wasn't technically a "divorce", but was an annulment -- a declaration that the marriage was never legally valid, rather than a divorce which ends a previously legal union.

Catherine of Aragon was originally married to Henry VIII's brother, Prince Arthur, who died after only a few months of marriage. The involved parties did not want to give up on the political advantages of their union, so it was proposed that she be married to Arthur's younger brother and the new English heir, Henry, though the marriage was delayed until Henry was older. However, marriage to your husband's brother was prohibited in church law as a form of incest. Therefore, in order for Catherine and Henry to legally marry, they had to argue that her first marriage to Arthur had never been valid. Catherine claimed that the marriage had never been consummated, and so the Pope granted the couple a dispensation acknowledging that Arthur had never been her legal husband, so she was free to marry his brother.

This became a problem when, many years later, Catherine had failed to provide Henry with a male heir. Although the English legal system would have allowed Henry and Catherine's daughter Mary to inherit (and, indeed, she later did become queen), Henry was set on having a son succeed him. Because he and Catherine had been married for years and had already produced a child, however, there was no way that he could argue their marriage had never been consummated. Instead, he wanted the new Pope to overrule the previous Pope's dispensation and declare that Arthur and Catherine's marriage had been valid, so he and Catherine were never legally married. However, Popes are generally reluctant to overrule the decisions of their predecessors because it is seen as a threat to papal authority. I'm sure there are Tudor experts on here who could go into more detail about why that particular Pope did not grant Henry's request (my expertise is on the church history side of things, not Tudor politics).

When Henry broke away and made himself the head of his own church, he was able to issue his own ecclesiastical ruling on whether or not Catherine and Arthur's marriage had been valid. So as you can see, the split between Catherine and Henry which led to the creation of the Church of England was not a "divorce" in the way that Wallis Simpson was divorced. Henry's other marriages ended in annulments (such as Anne of Cleves, on the ground of no consummation) or the deaths of his wives, under which conditions remarriage was allowed. As a two-time-divorcee with no annulment or dispensation, Wallis Simpson was considered to be a person with two living husbands, and so it was not permissible by Church of England law for the king to marry her.