These two modals express the speaker’s attitude about the verb. With should, the speaker is saying that it would be good if the verb happened; the attitude is one of recommendation, and not absolute, and the subject of the verb has a choice. With must the speaker is saying that there is no question about the necessity for the verb to happen; the attitude is absolute, and the subject of the verb, in the speaker’s mind, has no choice.
This is not a matter of attitude, but a neutral prediction that if something is going to happen, the verb’s happening is an essential requirement.
Here’s a passage from an article in the Korea Herald. Note should in the fourth sentence:
Currently, Korean women average 1.19 births throughout their lifetime, making Korea one of the nations with the lowest birthrate in the world. The fewer babies women give birth to, the few young people there are who engage in industrial production and the more old people who live on pensions. To meet the rise in welfare spending, the government should collect more taxes from youths, which will cut their consumption and make them less willing to work. Of course, the government’s fiscal stance will fall in chronic deficits due to excessive welfare expenditures for the aged.
The writer’s attitude of this passage is not one of admonition but prediction. The writer should therefore have written have to [actually, will have to] instead of should. (This is not the only mistake in this passage.)
In the class room the students are doing an activity. Student: “Mr. MacStein, do we have to look at the answer key after we answer the question?”
The speaker should use “should” or “do you want us to,” because the student wants to know Mr. MacStein’s attitude about “look.” A native speaker wouldn’t say have to in this situation, because looking at the answer key wouldn’t be considered a neutral essential requirement for something to happen. However, if the student wants to complain about looking at the answer key, have to would be proper, because it would mean “Is this the only way that we can do it?”