The example is using figurative language to describe the scene. Don't come out would be the normal form to use in that situation if one is simply listing the events, but the song is trying to evoke the sense of a struggle, a fight between the character and the words.

Don't is a more neutral term. It's a simple statement of fact - the words do not come. The line is about the singer's surprise that the words he's expecting don't come.

Won't personifies the words and grants them agency - they refuse to come out, while the character is trying to make them. The line is then about the singer fighting, trying to make the words come out.

Edited for clarity:

Won't does not always imply agency - it can also mean a prediction. To use an example in the comments, saying "The water won't drain" doesn't indicate that the water refuses to drain, it indicates a future tense. The implication of agency is largely contextual in this case, and related to the present tense of the line.

There, won't means "refuse to". Similar contexts can be found among the results here.

"won't" is a short form of "will not", where the verb will is used to express an ability, capability or an expectation:

Wood will float on water. Rock won't float on water.

The car will start when you turn the ignition on. I turn the ignition but the car won't start.

The lyrics you've cited express the the lack of the ability to speak, or an unfulfilled expectation that someone will speak when they open their mouth.

Using the future tense "won't" suggests that a rule or pattern of resistance to your intent has been established, so using "won't" is predictive that the trend will last into the future.

It's hinting at: "Even if I try again in the future, it still won't happen."

The future is used in the sense that it follows after your intention to make something happen.

You wouldn't use it if the situation was never to be repeated in the future.

He opens his mouth, but the words don't come out

This leaves it open that even though the words currently aren't coming out, then may in the future or when he tries a different approach.

He opens his mouth, but the words won't come out

This more strongly suggests that the problem is unfixable. No matter what he tries, the words won't come out. Not now, not in the future.

Note that this is not a stated fact, but simply a stronger implication compared to using "don't".

Secondly, keep in mind that this is a song. In lyrics, grammatical correctness can be overridden for lyrical flow. Especially for rap, easy pronunciation is essential to speaking quickly yet being easily understood.

Is it ok to say like this and is there any specific rule that says in which situations one can use won't when they would normally use don't/doesn't?

The distinction between "won't" and "don't" is effectively the speaker's opinion (in regards to how permanent this statement is). Logically, there is no rule that can define whether what the speaker believes is actually correct or not.

However, there is a guideline here. If you're writing a story in which the speaker is convinced the problem is unsolvable, "won't" is a better choice. If the speaker thinks that the problem is only temporary or an easy fix may exist, then "don't" is a better choice.

But neither options are definitive. It all hinges on how you want to portray the speaker.

"Won't" carries an implication that the thing not happening is desirable and plausible.

My dog doesn't talk. (implausible)

My dog doesn't bite. (undesirable)

My dog won't fetch my slippers. (plausible and desirable)

My dog doesn't fetch my slippers. (plausible and desirable -- doesn't is neutral on those questions)

EDIT

Roy makes an excellent point, in that "My dog won't bite" is also perfectly acceptable usage. I guess it's really more that "won't" implies some non-neutral desirability, either positive or negative, where "doesn't" -- well, "doesn't" necessarily do so. :)

Won't is used here instead of don't, to express the inability of the words to come out. It is not used in this case to express the future tense. I do not believe this use is ungrammatical.

Won't is of course a contraction of will not, and will has quite a lot of different uses in addition to its use to express the future tense

The Oxford English Dictionary gives 6 different uses of will as a modal verb: one of these matches the usage found here quite well:

Expressing facts about ability or capacity.

• ‘a rock so light that it will float on water’




• ‘your tank will hold about 26 gallons’

I would say that won't is generally used when you are expecting someone or something to do a certain action but it isn't doing it. But, doesn't is just stating a fact that someone or something isn't doing a certain action.

Won’t means it can, but it ‘currently’ will not.

Doesn’t means it does not.