We've eaten at both, more than once. Both have excellent beef, but I would say Shula's quality is a little bit better. However, Shula's accepts no discounts at all, and the sides are a la carte (in fairness, they are huge). Overall, attire is nicer at Shula's. Decor is darker. It can be very hard to get an ADR when a big convention is at the Swan/Dolphin.
Shula's also has one of the best desserts ever, the chocolate souffle.
Yachtsman is not casual, but the dress always seems less formal than Shula's (that's more of a WDW dress code enforcement issue, and 'that's all I have to say about that'.) Yachtsman meals usually have sides with them, the Yachtsman is on the dining plan, and at last visit, still accepts the discount formerly known as DDE.
It's usually a toss up for us where to go, but the 20% discount and the sides with the meal have made the difference last 2 trips.