This post explains the differences very well. They are the same in TypeScript as in JavaScript.
As for what you should use: You may define that on your own. You may use either, just be aware of the differences and it might make sense to be consistent.
The TypeScript coding style guide for the TypeScript source code (not an official “how to use TypeScript” guide) states that you should always use undefined and not null: Typescript Project Styleguide.