You can also fly or take a bus.
If you aren't planning on stopping along the way then you might as well take the catamaran....if you already have a car rented I would keep it until Split or if you wanted to stop for some days in between....but if it's just a one day thing I would take the catamaran.
If you want to stop on the way then rent a car. That is what we did. Just make sure that you account for parking cost which is high. We stayed at an Airbnb which included free parking. You should also be comfortable driving a manual transmission in a city and tight streets. Highway driving is easy.Edited: 17 October 2018, 13:31
The bus has the advantage that there are many more daily departures than the catamaran, so you can pick the time that works for you. (Unless you plan to sight-see along the way or will need a car for further travels after you finish in Split, can see no benefit to renting a car for this stretch.)
You can rent automatics if you don't drive manual. They are much more expensive though. There are so many places to stop between Dubrovnik and Split. Ston/Mali Ston, Makarska (or any of the little towns along the riviera...Tucepi, Baska Voda, Brela, Podgora, Omis, Solin....) As already stated if you don't want to stop anywhere then take the bus and choose the one that travels along the coast (Jadranska Magistrala) as opposed to the less scenic but faster highway A1. Personally I prefer the flexibility that a car offers. The catamaran would be a lovely trip as well. Tough choice! Happy planning :)