That's a big question, haha, they are incredibly different.
- For starter, Star Citizen universe has two parts to it. There's the single-player campaign called Squadron 42 and then there's the MMO part (Star Citizen)
- They both have their own physics system
- They have different approach on world-building. (Elite has procedurally generated planets, meanwhile Star Citizen is hand-crafted, it has fewer planets but more emphasis on detail)
There are so many differences, i don't think i can list all of them. In a way, Star Citizen and Elite is like Arma and Battlefield, they're both FPS, but each has different feels and gameplay. It really is a matter of preference.
If you want more detailed answer,
read these.
If you have more questions about the game, feel free to ask it on the Star Citizen thread.