"Have to" and "must" have the same meaning in the affirmative and interrogative forms when referring to obligation. Some grammarians think that "must" is slightly stronger, but for all practical purposes, they mean the same thing:

Doctors have to attend medical school for several years before they can practice medicine.

Doctors must attend medical school for several years before they can practice medicine.

While "have to" and "must" can be used interchangeably, there are differences in usage, as Michael Swan observes inPractical English Usage (Oxford University Press, 1995):

Both verbs can be used in British English to talk about obligation. (In American English, have to is the normal form.) British English often makes a distinction as follows. Must is used mostly to talk about the feelings and wishes of the speaker and hearer - for example, to give or ask for orders. Have (got) to is used mostly to talk about obligations that come from "outside" - for example from laws, regulations, agreements and other people's orders. Compare:

I must stop smoking. (I want to.)
I've got to [or I have to - Rachel]stop smoking. Doctor's orders.
This is a terrible party. We reallymust go home.
This is a lovely party, but we've got to [or we have to - Rachel] go home because of the baby-sitter.
..

Must you wear dirty old jeans all the time? (Is it personally important for you?)
Do you have to wear a tie at work?(Is there a regulation?)

The negative forms of "have to" and "must" carry very different meanings.

The negative forms of "have to" ("don't have to," "doesn't have to," "didn't have to") mean that an obligation is not necessary:

We don't have to go to work tomorrow; we can sleep until noon if we want! 

Marcy doesn't have to take any more English courses; she has satisfied the English requirement.

In times past, people didn't have to know how to use computers, but now they do.

In contrast, the negative of "must" ("must not") means that something is not permitted; it is prohibited.

In all cultures, people must not kill or steal.

You must not tell anyone about this. It is vital that the information be kept secret.

Children, you must not run into the street!

You could listen to the radio contains more personal involvement and subjectivity; You can listen to the radiois more strictly factual and objective. 

Similarly, How could I improve my English? is more a request for advice, whereas How can I improve my English?is more a factual question about available options. (But of course we can also answer this question by giving advice.)

Can and could, like the other modal verbs, have developed quite a range of meanings and uses. You ask how to explain the difference, and explanation can certainly help, but learning all the ins and outs of these verbs is a long process which requires plenty of experience, observation and experiment.

I think your question touches on two main issues: possibility vs. ability, and can vs. could.

Ability and possibility are similar ideas. If you've got the ability to do something, then it's possible for you to do it - in principle at least, although there might be something that prevents you. And, conversely, if you haven't got the ability to do something, then it isn't possible for you to do it. Both can and could (and other modals, especially may and might) are used to express various kinds of possibility, ability, permission and potential.

Could, of course, functions as the past tense of can, and like other past forms, it sometimes simply indicates past time: In those days there was no security and anybody could walk in, day or night.

But, like the past forms of other verbs, it can also indicate things like tentativeness, indirectness, deference and a wish not to impose. Compare:

I wanted to have a word with you (now)

I wondered if you needed any help (now)

with:

I want to have a word with you

I wonder if you need any help

The last two examples are more direct; the first two are more tentative, and possibly more polite – though that depends on other factors, too.

When we make suggestions, we often like to be rather tentative, so as to avoid giving the impression that we necessarily expect people to do as we say. Of course, it's possible to make much stronger suggestions – e.g. I think you should listen to the radio – but we generally prefer to adopt a more 'take it or leave it' approach, and lessen the possibility of offence on either side. So that's why You could ..... is so commonly used in suggestions (and, by the same token, Could you .....? is often used for requests). In fact, it's so common that it makes sense to learn it as one of the standard formulae for giving advice and making suggestions (and to reserve You can ....., in similar contexts, for factual statements). And, by the way, it's quite common to add always: Well, you could always listen to the radio. Of course, the well at the beginning isn't necessary, but it also contributes to the general impression of tentativeness.   

The Rules of Regular and Irregular Plurals

An idiom is a word, phrase, or expression that cannot be taken literally because the meaning is figurative. Idioms are a form of expression peculiar to a language. The English language has many idioms that mention food. The following sections provide definitions of some common English idioms and proverbs that include fruit, vegetables, and baked goods.

Fruit Idioms

The following English idioms figuratively use some sort of fruit, usually to describe something negative but occasionally something positive.

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Tongue-Twisters

A tongue-twister is a sequence of words that is difficult to pronounce quickly and correctly. Even native English speakers find the tongue-twisters on this page difficult to say quickly. Try them yourself. Try to say them as fast as possible, but correctly!