Enemmän is also more, but it's the comparative form of paljon, a lot.
Here's how to use them in sentences:
- Saisinko lisää kahvia? - Could I have more coffee, please? (I already had some but I drank it and now I want more.)
- Saisinko enemmän kahvia? - Could I have more coffee, please? (You only gave me some and I'd like to have some more.)
- Ota lisää ruokaa. - Take more food. (Your plate is empty. Take more.)
- Ota enemmän ruokaa. - Take more food. (You took too little. Take more.)