What's the difference between writing a will and the nominee form that I am supposed to pass to my insurance agent, don't they essentially do the same thing?

Essentially yes.

The difference lies in that there are two things in Singapore law that a will won't cover.
1) Insurance policies
2) CPF (And you wonder why CPF works like insurance.)

That's why you have to fill up a nominee for your insurance polices and CPF, and a will for every other thing.