How does Buddhist (supra-mundane) Happiness differ from ‘mundane’ happiness?

Nibbana (the only true reliable perfect happiness) does not come with a price. It is 100% free.

Therefore, the difference between the two forms of happiness is: (i) Nibbana is completely free; (ii) where as the worldly pleasure of sensuality, materiality & status is extremely financially expensive.

If there is the wrong view (miccha ditthi) that sensual & worldly pleasures are being 'sacrified' at a 'price', the hindrance of lust will be difficult to overcome.

Friend Ariṭṭha, do not say so. Do not misrepresent the Blessed One; it is not good to misrepresent the Blessed One. The Blessed One would not speak thus. For in many ways the Blessed One has stated how obstructive things are obstructions, and how they are able to obstruct one who engages in them. The Blessed One has stated that sensual pleasures provide little gratification, much suffering and despair, and that the danger in them is still more. With the simile of the skeleton…with the simile of the piece of meat…with the simile of the grass torch…with the simile of the pit of coals…with the simile of the dream…with the simile of the borrowed goods…with the simile of fruits on a tree…with the simile of the butcher’s knife and block…with the simile of the sword stake…with the simile of the snake’s head, the Blessed One has stated that sensual pleasures provide little gratification, much suffering and despair, and that the danger in them is still more.

https://suttacentral.net/en/mn22

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So too, Māgandiya, formerly when I lived the home life, I enjoyed myself, provided and endowed with the five cords of sensual pleasure: with forms cognizable by the eye…with tangibles cognizable by the body that are wished for, desired, agreeable, and likeable, connected with sensual desire and provocative of lust. On a later occasion, having understood as they actually are the gratification, the danger, and the escape in the case of sensual pleasures, I abandoned craving for sensual pleasures, I removed fever for sensual pleasures, and I abide without thirst, with a mind inwardly at peace. I see other beings who are not free from lust for sensual pleasures being devoured by craving for sensual pleasures, burning with fever for sensual pleasures, indulging in sensual pleasures, and I do not envy them nor do I delight therein.

https://suttacentral.net/en/mn75

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As I abided thus, diligent, ardent, and resolute, a thought of sensual desire arose in me. I understood thus: ‘This thought of sensual desire has arisen in me. This leads to my own affliction, to others’ affliction, and to the affliction of both; it obstructs wisdom, causes difficulties, and leads away from Nibbāna.’ When I considered: ‘This leads to my own affliction,’ it subsided in me; when I considered: ‘This leads to others’ affliction,’ it subsided in me; when I considered: ‘This leads to the affliction of both,’ it subsided in me; when I considered: ‘This obstructs wisdom, causes difficulties, and leads away from Nibbāna,’ it subsided in me. Whenever a thought of sensual desire arose in me, I abandoned it, removed it, did away with it.

https://suttacentral.net/en/mn19

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As a mighty flood sweeps away the sleeping village, so death carries away the person of distracted mind who only plucks the flowers (of pleasure). The Destroyer brings under his sway the person of distracted mind who, insatiate in sense desires, only plucks the flowers (of pleasure). The fool worries, thinking, “I have sons, I have wealth.” Indeed, when he himself is not his own, whence are sons, whence is wealth? A fool who knows his foolishness is wise at least to that extent, but a fool who thinks himself wise is a fool indeed. Though all his life a fool associates with a wise man, he no more comprehends the Truth than a spoon tastes the flavor of the soup. One is the quest for worldly gain, and quite another is the path to Nibbana. Clearly understanding this, let not the monk, the disciple of the Buddha, be carried away by worldly acclaim, but develop detachment instead. The good renounce (attachment for) everything. The virtuous do not prattle with a yearning for pleasures. The wise show no elation or depression when touched by happiness or sorrow. He is indeed virtuous, wise, and righteous who neither for his own sake nor for the sake of another (does any wrong), who does not crave for sons, wealth, or kingdom, and does not desire success by unjust means. Whoever is overcome by this wretched and sticky craving, his sorrows grow like grass after the rains.Beset by craving, people run about like an entrapped hare. Held fast by mental fetters, they come to suffering again and again for a long time.

https://suttacentral.net/en/dhp#2

Here is an extract from Me and Mine: Selected Essays of Bhikkhu Buddhadasa:

An equally close connection pertains between looking within and true "happiness." In the Buddhist texts there are two kinds of happiness. One of these is found in home life, called gehanissitasukka, the kind of happiness that derives from being a householder and raising a family. Contrasted with outward happiness is nekkhammanissitasukka, literally the "happiness that comes from forsaking home life." The reference is to a mental forsaking, a state of mind in which there is no longer the idea of "my home." This kind of happiness comes from looking within, observing things according to the dhamma.

The happiness of home life is called lokiya sukha or worldly happiness; and the happiness that goes beyond home life is called lokuttara sukkha or transcendent happiness. It all depends on the state of mind. If a person's mind is worldly, he can stay in a monastery or in the forest but will attain nothing more than the happiness of home life. Solitude in a monastery or any other place cannot guarantee lokuttara sukha. This happiness depends on the ability to look within.

No matter where we are, we have the power to transcend a preoccupation with the world, above and beyond the "home", simply by looking within. There is a large, guarantee return. Without any outlay of capital we obtain this special happiness which appreciates continuously. As the Buddha said, "Nibbana costs nothing; we pay nothing for it." All we have to do is "throw away". If we will just throw everything away then we will get nibbana. This means simply having a mind high enough not to get stuck in the world. That is all there is to it. We throw away the world completely and get nibbana in return. We do not have to invest anything. We only have to be empty, to live rightly, and nibanna will come of itself.

Note:

• It says that the Buddha says, "Nibbana costs nothing; we pay nothing for it." -- I don't know which sutta the venerable is paraphrasing when he says that, but that agrees with Dhammadhatu's answer
• Being required to "throw everything away" everything might seem like a cost; on the other hand the one example of what you have to throw away is "the idea of 'my home'" -- if all it costs you is an idea (e.g. "ego") that's maybe not a really tangible cost.
• You wrote "supra-mundane" in the OP and I'm pretty sure that the Pali word for that is lokuttara (similarly, sukha is "happiness").

Lokuttara is used to refer to the four paths and four path-fruitions.

Apparently its etymology is,

Loka + Uttara = Lokuttara. Here Loka, means the five aggregates. Uttara means above, beyond or that which transcends. It is the supra mundane consciousness that enables one to transcend this world of mind body.

That meaning (i.e. "above") might be why Bhikkhu Buddhadasa wrote, "having a mind high enough not to get stuck in the world".

Bikkhu Buddhadasa expands on that in this lecture:

Even an old man who can hardly move about and must remain at home all the time, if he knows Dhamma at this level, while still living in the home, may attain the happiness that comes from forsaking the home life. This is because the term "forsaking the home life" refers to a mental forsaking to a state in which the mind transcends worldliness and goes beyond it. A person who is living at home may experience the happiness that comes from the home life. Or, he may experience the happiness that comes from forsaking the home life, provided he is capable of looking within using the technique and method of Dhamma.

Reading that, maybe you could say that another "cost" (or maybe "prerequisite") is "knowing Dhamma": i.e. time, effort, attention required to learn and to practice the Dhamma.

This answer doesn't directly address the sentence, "The happiness that can be found within our day to day life has to be sacrificed in the form of sense-restraint if one becomes a disciple of the Buddha" in the OP, but it does try to illuminate what was asked about i.e. supra-mundane happiness and cost.