Corinna,
I have completed eight Disney Cruises in the past five years so I'm not sure if my expectations where influenced by my dedication to DCL but I did go my Anthem cruise with an open mind and a desire to have fun. That being said the price for my Anthem cruise was far less than a Disney 7 day cruise so that was great. Also the Anthem is a beautiful ship and they are plenty of activities on the ship for everyone that DCL doesn't do/have such as bumper cars, rock climbing, virtual skydiving, roller skating, wave machine, casino, trapeze fun and north star viewing, etc. I also like the technology around the ship such as the virtual/interactive hallway maps, robot bar but my favorite was the Two70 room with virtual wall (in the evening) and robotic HD screens. RCCL does a great job in managing the thousands of people around the ship with all these activities that it never felt too crowded.
I also took advantage of some of the paid specialty restaurants by dinning at Wonderland, Chef's Table @ Prime, Jamie's Italian, Michael's Pub and even Johnny Rockets. The food at the specialty restaurants where good and my favorites where the Chef's table and Jamie's Italian. Wonderland is a unique experience and the food was good but I was probably never visit that restaurant again. RCCL has copied DCL and does the rotational dining which I enjoy but the food was nothing special and IMHO below DCL as far as taste. I also thought that the Windjammer had the same menu/spread everyday during lunch which was discouraging. I'm not a soda drinker but if you are or have children that do like soda it's going to cost you. The tea and lemonade dispenser is syrup mixed with water and I either got too much syrup or too much water.
The shows on the Anthem where horrible and after one of the shows (The Gift) which by the way had wonderful props I asked my friend sitting next to me what did he think? His response was "What the hell did I just watch" which was my feelings as well. The same was true of the Rock of Ages which was discouraging because I am a Queen fan. The cast where great and are talented but it's the stories for these shows that is lacking.
The service was far below my experiences with Disney and it also seemed like the crew was not happy or didn't get along with each other and it was clear something was wrong but I couldn't put my finger on it. I also had one issue in the dining room, spa and shore excursion desk that Disney would probably wouldn't let happen or would have resolved immediately and not try to send me off to someone else. I also didn't like the constant sales push and big sales in the Royal Esplanade area, the numerous jewelry shopping presentations on the stateroom television. Speaking of the stateroom television the channel selection is bad and they charge for movies so DCL is clearly better.
I'm going to end here and just say that my reasons for sticking with Disney are simply the service, crew and the shows. If you have any specific questions I didn't answer please PM me I will be more than happy to do my best to answer them.