There is no difference between relevance and relevancy, though relevance is preferred in all varieties of English (though relevancy is the older form). Relevance is about ten times as common as relevancy in current U.S. news sources, and the gap is even wider in British, Australian, and Canadian sources.
Examples
Relevancy does occasionally appear in edited publications—for example:
At a time when public schools are battling for relevancy in high school sports, there are always those special moments … [Los Angeles Times]
This striking production leaves a lingering impression of that curse and its relevancy. [Irish Times]
Relevancy of the message and content are paramount. [Forbes]
Libraries have certainly morphed in the last few years as they fight for relevancy in an ever-digital culture. [Globe and Mail]
Relevancy is not wrong in these cases, but the shorter and more common relevance would work in its place.
Ngram
Below is an Ngram charting the words’ use in books published from 1800 to 2000. It shows that there is no competition—relevance is the word.
