Major Contribution(s) to Occult Studies:
The Enochian language and Enochian magical system, including the Enochian keys.
Dee, through the use of a "shewstone" and mediums, most notably Edward Kelley, communicated with a series of angels who revealed the Enochian language, magic and rituals to him. The name Enochian refers to the Biblical figure Enoch, whom Dee claimed had been the last human to understand the language.
This is in stark contrast to other occultists of his day, who generally viewed their practices as being primarily spiritual in nature rather than practical.
A variety of modern occult groups use portions of Enochian magic, including the Golden Dawn, Thelema, and LaVeyan Satanism.
Contributions to Science:
Dee was an accomplished mathematician and taught it at university. He was well versed in Copernican models of the universe, although historians debate whether Dee accepted Copernicus's earth-centric models as physical fact.
Dee was also an experienced astronomer, as detailed knowledge of the movement of the heavens was central to astrology.