Alys is the English spelling of the name, historically. (We Welsh borrowed it thus from the neighbours!) It's from the Old French Aliz, and as The Oxford Dictionary of English Christian Names says "By the 12thC ... it was a very common name in England as well as France ... In England it was usually Alys...In the middle of the 17th C it was regarded as an old-fashioned, country name and fell into general disuse, until its revival in the middle of the 19thC by the romance writers, often in the latinized form Alicia. CMY says of it (1863) 'now with constantia and Edith a favourite fancy name'".
So historically, Alice is the made-up, fancy name, and Alys the traditional English spelling. Alys was good enough for Chaucer, certainly - it's the name of the Wife of Bath in the Canterbury Tales.