As mentioned, the dinosaur exhibit is amazing. I probably spent hours in there. An impressive fraction of the bones on display are real, not casts. In addition, all the touchscreens have interesting information about the dinosaurs, like misconceptions and new discoveries.
The gem room is also full of great exhibits and information. I liked that they displayed carved pieces as well as the typical rocks and crystals. By the time you get to the room with the seemingly endless cases and mirrors, it almost feels too overwhelming. But I can't really complain about there being too many gems!
I have a really soft spot for the wildlife hall. The person who posed the animals seems to have a sense of humor. The lion attacking the man on camel: awesome. One of the bear exhibits is very cleverly posed, with part of the scene in the wall and the male bear in the middle of the hall.
The only con is that the Arctic and Egyptian rooms are on the top floor. By then, I'm too tired to appreciate them. It may be best to see those first and then make your way down to the dinosaurs. Also, the bird hall is a little strange, with some birds posed upright and others left lying in a very dead posture with their feet tied together. You may find this either amusing or creepy.
Remember to get there early and plan for a full day if you want to see both the Art and Natural History museums!