While I haven't stayed at the Courtyard, there's a very big difference between the hotels. The Courtyard is the King Kamehameha hotel, which was independent until recently remodeling and aligning with Marriott. It was built in the 1970's in the heart of Kailua-Kona, so is walkable to restaurants, shopping, etc. The rooms look nice and we've had a great time at the luau at the hotel even when not staying there. Don't expect a huge resort environment - there's a pool and small neighboring beach, but it is not a mega-resort like most of the Big Island hotels.

The Sheraton is also a renovation, but is in a development south of Kailua-Kona. While it is still somewhat similar to an urban hotel in design, they have much more of a resort environment, with well kept grounds and far larger pool areas, etc. for enjoyment. The downside here is that there is very limited dining, etc. close by, but it's not difficult to get into Kailua-Kona if you want something different.

I think it really comes down to what you're looking for in a vacation. If you're out exploring the island every day, I think either is fine, leaning to the Courtyard if you want convenience to non-hotel restaurants. If you're looking for a lounge by the pool type experience, the Sheraton is far better (and the cabanas are a great deal, especially if you're a Platinum)!