We know that Easter must always occur on a Sunday, because Sunday was the day of Christ's Resurrection. But why the paschal full moon? Because that was the date of Passover in the Jewish calendar, and the Last Supper (Holy Thursday) occurred on the Passover. Therefore, Easter was the Sunday after Passover.
The Church does not use the exact date of the paschal full moon but an approximation, because the paschal full moon can fall on different days in different time zones, which would mean that the date of Easter would be different depending on which time zone you live in. For calculation purposes, the full moon is always set at the 14th day of the lunar month (the lunar month begins with the new moon). Likewise, the Church sets the date of the vernal equinox at March 21, even though it can occur on March 20. Both approximations allow the Church to set a universal date for Easter.
Still, Easter isn't celebrated universally on that date. While Western Christians use the Gregorian calendar (the calendar that's used throughout the West today, in both the secular and religious worlds) to calculate the date of Easter, the Eastern Orthodox continue to use the older, astronomically inaccurate Julian calendar. Currently, March 21 on the Julian calendar falls on April 3 in the Gregorian calendar. Therefore, for the Orthodox, the Sunday following the 14th day of the paschal full moon has to fall after April 3, hence the discrepancy in the date of Easter.