The coronation of Charles gave the Pope the power to excommunicate any person who refused to bear arms in any cause of Charles, thereby in effect giving the Vatican control over all the armies of any Christian nation. This is the ultimate example of church/state affiliation, because the individual conflicts of two countries, whether border disputes, questions of succession, or any other problems, immediately conflated into a continent-wide, Pope-driven war, and the spiritual integrity of Pope Leo was set aside in favor of the very earth-bound desires of the powers of the Vatican, itself often orchestrated by powerful Italian families.
The crowning of Charlemagne by Pope Leo III was important in a number of ways.
For Charlemagne, it was important because it helped to give him greater credibility. It gave him the status of an emperor, making him the only emperor in Europe west of the Byzantine emperor in Constantinople. This made Charlemagne seem like the most important ruler in Europe.
For the Pope, it was important as well. By crowning Charlemagne, Pope Leo got himself an important ally. Leo had had to flee to Charlemagne's court just the year before when a rebellion erupted against him. By crowning Charlemagne, Leo was connecting himself politically to Charlemagne and thus gaining some power himself.
Finally, the crowning of Charlemagne was important for Europe as a whole. It represented the creation of the Holy Roman Empire. This was a first step towards creating a true European society such as had not existed since the "fall" of Rome.