Capturing Groudon and Kyogre
In Ruby and Sapphire, Groudon and Kyogre were respectively available earlier in the Cave of Origins, in Emerald they're both available, but not until after the Elite 4 and are in game-exclusive locations. The caves' entrances don't both appear at the same time, they rotate until until one is captured/defeated and then only one will be around.
To find out where one is, go to the Weather Institute on Route 119 and a man there will explain that strange weather is occuring on a route. The entrances' times are limited, if you fail to get there in time it will move elsewhere.
Terra Cave is home to Groudon, and the entrance roams between Routes 114, 115, 116, and 118.
Marine Cave is home to Kyogre, and the entrance roams between Routes 105, 125, 127, and 129. Where it is, it will be accessible by a unique dive spot that is not usually on the route, so Dive is required to access this cave.
Other extras in Emerald
Also after you defeat the Elite Four, a new underground cave will appear above Route 114, in the back of the Fossil Maniac's house. In this cave, you can catch a Ditto as well as pick up the Root Fossil or Claw Fossil - whichever one you didn't get back in the Mirage Tower, meaning both Anorith and Lileep are available in Emerald Version.
If you complete the Hoenn Pokédex (all 200-Jirachi and Deoxys aren't required), Professor Birch in Littleroot Town will give you a starter from the Generation II games of your choice.
Note that this is the only point in any of the five Generation III Pokémon games that you can get these Pokémon.