Why does the Easter date vary so much?

In the Western church (Catholic and Protestant churches) Easter occurs on the first Sunday after the first full moon after the Spring Equinox. Spring equinox occurs around March 21 but can occur a day earlier. For the church's purposes the equinox occurs March 21. The full moon can occur from March 21 and up to 29 days later.

The Eastern* church (Orthodox church) uses a similar formula but follows the older Julian calendar, and defines certain days. Thus "full moon" occurs on the 14th day of the month, not when the moon is full.

Jesus died at Passover. Passover begins Nisan 14† which is during the first month of the Jewish year. The Jewish calendar is lunar-solar which means that the months all begin at the new moon. Months are 29 or 30 days long. Because there are more than 12 lunar months in a year the Jewish year occasionally has an extra month. The Jewish New Year starts about the time of the spring equinox (northern hemisphere).

So Jesus was crucified on a Friday during a full moon. The Easter date is designed to have Good Friday and Easter Sunday land close to the first full moon after the spring equinox (which approximates the Jewish New Year) while maintaining the days of the week so that every year the crucifixion and resurrection days land on Friday and Sunday respectively.

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*Some Eastern churches use the Western date.
†Modern Jews use Nisan 15.