Here is the basic difference, native speaker-wise:
As found in orders or instructions, in formal language, including technical language. must is to have to do, but more formal.
- All students must enter the gym through the side door this week.
- Lever B must be upright before the door is closed.
- Applications must be filed before June 2nd.
Shall is no longer used in sentences like the ones above to signal an order or instructions.
Must here in French is devoir or Il faut que etc.
As found in everyday language where one person tells another they have to do something. To have to do something = semantically, must do something.
You must go back to the office now or your boss will find out your are out.
You have to go back to the office now or your boss will find out you are out.
We must finish this job by 5 o'clock.
We have to finish this job by 5 o'clock.
He must do it immediately, mustn't he?
He has to do it immediately, doesn't he? [has got to do it, hasn't he]
Must you keep doing that?
Do you have to keep doing that?
In this sense, must and shall are not related.
past tense: He must have seen the dog run out the door. [devoir avoir + participe passé]
In legal language, shall is still used to express a binding obligation in a contract or treaty or those kinds of documents and contexts, and must would never be used there in a contract or treaty. Must is not much used in contracts, though, of course, it might be. That said, shall in contract means must in everyday language:
- The parties shall meet once a week for the first six months of operation.
Here is an example of a treaty between the US and Turkey where it is used 58 times.
international investment treaty
It is discouraged now by some legal writers. In any event, it just means will or imposes an obligation, depending on context, and here is a technical legal discussion about that, which is not super relevant here:
shall, will and consistent legal writing [French documents usually use the present tense even when expressing obligation]
Everyday language: Traditionally, an intention to do something was usually expressed by: shall and will, shall for the 1st person singular and plural, and will for the other person**s. And in that sense, will can replace all shalls.
Please note: The Brits do tend to use it quite a bit this way, and in general, "Shall I" or "Shall we"" is used to offer to do something for someone in both "BrE" and "AmE".
I shall [intend to leave] leave at 2 o'clock today unlike other days. They will leave [intend to leave] at 5 o'clock. [future in French and has the same feel of formality.]
Shall I leave the door open when I leave? [Yes, please do, No, etc.] [Voulez-vous que je laisee, etc. pour rendre service]
This last use of shall to offer to do something is very idiomatic and the answers to it vary and are also idiomatic:
- Shall we go now? Yes, let's do.
- Shall I open the window? Yes, please do.
In AmE, will has mostly replaced any use of shall in everyday conversation (except the usage given above). But not necessarily in formal contexts. ["Shall we leave it at that?" Also, the contracted forms are very used: You'll, we'll, etc. "Shall" is no longer used much to express an obligation in everyday speech:
- You shall arrive on time every day. [that would be an order and is old-fashioned]
-Must expresses an obligation in everyday usage.
shall is a special case in legal language. (consistency rule: shall versus will)
shall is either first person singular/plural for future intention in everyday usage: I shall do that work later.) Often used to offer to do something for someone or to make a suggestion about a situation: Shall we leave now? where "will" cannot be used.
That use of shall in French is future or expressed as a future.
In everyday language, there should be no reason to confuse the use of must (obligation) and shall (suggestion or offer to do something, used in the first person singular and plural, mostly as a question).
- You must leave now! = Il faut que tu partes maintenant.
- Shall I open the window? = Voulez-vous que j'ouvre la fenêtre?
- Shall we go? = Partons(-nous)?
- You shall do it now. [désuet, en anglais] = Tu le finiras.
- You shall do it now. an order, in modern speech has become: You will do it now.
The only confusion between the two might possibly arise in legal language because in everyday language, the old-fashioned use of shall as in: "I say he shall go tomorrow"!, an order, which is the same as: I say he must go tomorrow!, is no longer commonly used in English. Even when expressing intention, will has mostly replaced shall ("I shall do it tomorrow" has become "I'll do it tomorrow") except for certain speakers who seem to love their "I shalls": "We shall finish this discussion in the morning. It's late." And that in French is a future tense.