ませんか invites, ましょう proposes something already agreed on. The negative question is the politer of the two, which is counter-intuitive at first.
Also known asmashoumasenkaましょう ませんかlets in japaneseinvitation
V-ませんか invites the listener to do something together: “Won't you …?”
いっしょに きょうとへ いきませんか。
Won't you go to Kyoto with me?
Although it looks like a negative question, V-ませんか is a polite, soft invitation — “shall we / won't you …?”.
In the textbooksMinna I, lesson 6Genki I, lesson 3
V-ましょう proposes doing something together: “Let's …”.
ちょっと やすみましょう。
Let's take a short break.
ましょう expresses “let's”. It is also the natural reply when accepting a ませんか invitation: 行きませんか → ええ、行きましょう.
In the textbooksMinna I, lesson 6Genki I, lesson 5
Besides “shall we …?”, V-ましょうか offers to do something FOR someone: “Shall I …? / Let me …”.
にもつを もちましょうか。
Shall I carry your bag?
Shall I do it?
Same form as the “let's” ましょうか, but here the speaker alone performs the action as a favour — often with (わたしが).
Compare ませんか, which invites the listener to join you, and ましょう, which proposes doing it together.
In the textbooksGenki I, lesson 6