According to the Congressional Record Service the "with distinction" is just a higher degree of the medal:
Presidential Medal of Freedom. The Presidential Medal of Freedom is considered the highest civilian award of the United States government. President Truman first awarded the Medal of Freedom to reward war-connected acts or services during World War II. It was later re-established by President Kennedy in Executive Order 11085 CRS-4 of February 22, 1963, to recognize persons who have made especially meritorious contributions to the security or national interests of the United States; to world peace; or to cultural or other significant public or private endeavors. There are two degrees of the Medal, the higher being the Presidential Medal of Freedom with distinction.
Also the insignia is a little different:
The badge of the Presidential Medal of Freedom is in the form of a golden star with white enamel, with a red enamel pentagon behind it; the central disc bears thirteen gold stars on a blue enamel background (taken from the Great Seal of the United States) within a golden ring. Golden American bald eagles with spread wings stand between the points of the star. It is worn around the neck on a blue ribbon with white edge stripes.
A special, rarely given grade of the medal, known as the Presidential Medal of Freedom with Distinction, has a larger execution of the same medal design worn as a star on the left chest along with a sash over the right shoulder (similar to how the insignia of a Grand Cross is worn), with its rosette (blue with white edge, bearing the central disc of the medal at its center) resting on the left hip. When the medal With Distinction is awarded, the star may be presented descending from a neck ribbon and can be identified by its larger size than the standard medal (compare size of medals in pictures below; President Reagan's was awarded With Distinction).