I believe it might harken back to the fact that the Royal family provided much of life's requirements to their people. The crest is actually the crest of the Earl of Galloway. The Stewart Clan has no Chief so his is used since he is the closest there is to a Clan Chief. The motto for the Clan is "Virescit vulnere virtus (Courage grows strong at a wound)". Also, the two spellings come from the fact that when Mary. Queen of Scots was the head of the Royal family here second husband provided a strong French influence. The name Stewart was difficult for the French to pronounce due to the spelling so it was changed to Stuart to make it easier on the French. That spelling continued on after her with James IV. The Stewart Clan is also broken down into many different Clans such as Atholl, Galloway, and Ardvolich among many others.
Just an FYI thing.