I'd skip Animal Kingdom and not think twice about it. First, I much prefer the old Hollywood theme of DHS to the theme at Animal Kingdom. Animal Kingdom is no zoo. Sure, there are animal, but it's not like a zoo where you think to yourself, "you know, I want to go see the giraffes" and then you head off to the giraffe exhibit. You can see a number of animals on the safari. You can do the gorilla trail to see gorillas and hippos, but no other animals on the trail interest us (birds and fish... meh). You can see tigers on maharaja jungle trek, but no other animals really interest us there (more birds, I think a komodoa dragon and some bats). So really, as far as animals go, I see them on the safari which means I am constantly moving and can't stay as long as the exhibits I want and skip the ones I don't. I'd MUCH rather go to one of the 3 zoos around here that are all within a 45 minute and see what I want, when I want, for however long I want. So, for theme, DHS wins.
That brings me to the rides. In my opinion, Animal Kingdom doesn't compare to DHS rides-wise. Expedition Everest at Animal Kingdom is one of the best themed rides in the park and is a fun roller coaster. at DHS, Tower of Terror is right up there with Expedition Everest as far as amazing theming goes (and, in my opinion, is even slightly better) and a fun ride. Rock n Roller Coaster, also at DHS, is a fantastic roller coaster that I find to be more fun to Everest. So, I think that you get 2 rides at DHS to compare to only 1 at Animal Kingdom. Dinosaur at Animal Kingdom is jarring and not as much fun. When you compare that to Star Tours at DHS, well, I don't think you can compare the two. Star Tours also has the added benefit of having multiple experiences so you can do it many times and have a different ride each time. Primeval Whirl is a good family ride at Animal Kingdom, but Toy Story Midway Mania is enjoyed more by our family and doesn't have quite the same motion sickness result that Primeval Whirl causes. Great Movie Ride at DHS is my favorite ride and I also love the queue. So, for rides, DHS wins.
Regarding shows. Festival of the Lion King is a fantastic show at Animal Kingdom. No doubt about it. Fantasmic at DHS is my favorite show, though, so if I have to compare the best show in my opinion at Animal Kingdom (FOTLK) to the best at DHS (Fantasmic), DHS wins out there. I don't like Finding Nemo the Musical (AK) nearly as much as Little Mermaid (DHS). Flights of Wonder at Animal Kingdom is just meh. We have bird shows at all of the zoos around here. I've seen Beauty and the Beast Live on Stage at DHS at least a dozen times and still enjoy it. My six-year-old daughter loves Indiana Jones Epic Stunt Spectacular along with the rest of the family. So, for shows, DHS wins.
When it comes to dining, it gets a little more complicated. Counter service dining at DHS is lacking. The new Pizza Rizzo has a fun theme so I think it is better than Pizzafari at Animal Kingdom because I find that theme to be lacking. I also like seeing all of the movie artifacts at Backlot Express at DHS, but I also enjoy the theme of Restaurantosaurus at Animal Kingdom. That has to be a wash. I don't really think any of the other counter service restaurants at DHS are worth writing home about. At Animal Kingdom, I would recommend Yak and Yeti counter service along with Restaurantosaurus. So, both parks have a couple of decent counter service restaurants and a few that are just meh. Definitely a wash there. At DHS, I think that they have the 2 best themed table service restaurants in Sci Fi and 50's Prime Time (and I'd put them in that order but love eating at both). I think that the themes at those restaurants are worth eating there, regardless of the food. But, make no mistake. The food, especially at 50's Prime Time Cafe is quite good. For little ones, Hollywood and Vine is a great way to meet Disney Junior characters or Mickey and friends, but the food is average at best. When you compare that to Tusker House at Animal Kingdom, where you can also meet Mickey and friends (it is hosted by Donald) and the food is exotic and good, I have to give Tusker House the edge. Unfortunately, I don't think that the other restaurants are all that great here. As Landry's Club members, I really wanted to like Yak and Yeti as well as Rainforest Cafe. It's strange because I enjoy all of the Rainforest Cafe restaurants I've gone to except the Disney ones. The menu is limited and I don't think the food is as flavorful as at other restaurants. Yak and Yeti, sadly, was also lacking the flavor we were hoping for. I'm not saying that it was bad. Not at all. It just didn't have enough flavor. Anyway, for dining, I'm going to give the edge to DHS.
So, yeah. DHS wins in atmosphere/theme, rides, shows, and dining. Then, when you add in the location and how it feels like it takes forever to get to Animal Kingdom from any resort except Animal Kingdom Lodge and the All Stars and it's a clean sweep for DHS.