When steam engines were fully accepted as suitable engines for ships, the paddlewheel was the preferred form of propulsion for decades. Many people suspected that the propeller would be a better mode, but many preferred the paddlewheel. The paddlewheel had the obvious disadvantage that it was vulnerable to damage by cannonfire, but still had its proponents. Finally, in 1845 the British Navy decided to compare the two systems once and for all. For the comparison, two roughly equal ships were selected. In the paddlewheel corner was the champion, HMS Alecto, weighing in at 880 tons, with a 200 hp steam engine. The challenger was the propeller driven HMS Rattler, also weighing in at 880 tons, and also with a 200 hp steam engine.
Over an 80 mile course, the Rattler easily won a one-on-one race with Alecto, by 23.5 minutes. In a 60 miles race, using a different propeller design and against strong winds and seas, Rattler won by 40 minutes. Twelve heats were held in all. The final and most exciting round, although largely a publicity stunt, had the two ships cabled together stern to stern, for a full power tug-of-war.
The Alecto reached full power quicker, and pulled the Rattler backwards for 5 minutes. However, at five minutes the Rattler reached full steam, and brought the Alecto to a stop. With the Alecto's paddlewheels turning furiously, the Rattler then towed the Alecto backwards at a rate of 2.8 knots. From then on all new ships of the British Navy were propeller driven.
This site has a great contemporaneous record of the contest.