Suggest an activity politely.
The rule
The form in ましょう proposes to do something together. ませんか invites in the form of a question and often leaves more room for the answer.
Examples
いっしょにいきましょう。
Let's go together.
おちゃをのみませんか。
Would you like some tea?
Remember
- ましょう proposes; ませんか formulates an interrogative invitation.
- Suggest an activity politely.
Common mistake
Understand ませんか as a simple negative statement: context makes it an invitation.