Pokemon Cosmic Emerald is the same in terms of story as Emerald. Nothing has changed in terms of the characters and events in the main storyline. Just play through it like you would Emerald.

The story may be the same, but the gameplay has been improved. One thing is that the PRNG glitch has been fixed. The PRNG glitch was notorious back in Emerald, but it has been fixed for Cosmic Emerald.

There are also a lot more Legendary Pokemon that you can catch in the game. Certain events are also made available in the game and depending on if it is night or day will determine the event. There are also individual patches you can do to the game.

All Generation I to III Pokemon are available in the game
Use an Evo Stone to evolve Pokemon that evolve only through trading
A lot more Legendary Pokemon to capture
Poke Balls can be reused when capturing has failed
TMs can be used in an unlimited amount
Pokemon gets experience whenever enemy Pokemon is caught
PRNG has been fixed
You can run indoors
EVs will max out at 252
The game uses Black and White’s Repel system
Day and night cycle
Special events specific to day or night

If you have any questions, please let me know!
Thanks for checking out my listing.