"Stuart," came into being through Mary Queen of Scots, who married into the French royal family. I believe, I may be wrong, that there is no "W," in the French language, so they changed the spelling to "Stuart."

"Stewart," came to be the name of the Scots royal family through the marriage of Marjory Bruce, the daughter of Robert The Bruce, to Walter - High Steward of Scotland. Thus "Steward," through time became Stewart.

""Stuart," is the wrong spelling.