Do they both play a role in getting rid of the hindrances or is one better then the other? How do they eliminate them?
To your question, “Which eliminate the hindrance permanently, Noble Eightfold Path or supramundane?” …. There are two level to this Path – the Mundane and the Supramundane. It is at the higher level of this Path that the Hinderances are eliminated permanently, but it happens in stages.
As aforementioned, ascertaining the truth has two parts, i.e; mundane truth and supramundane truth. Ascertaining the truth in matters superficially is known as mundane truth and at this level, the five hindrance remains. This mundane truth would lead to the supramundane ultimately. The ultimate meaning of samma dhitti (right view) is ascertaining the supramundane truth.
What involves in the category of supramundane truth that would help eradicate the five hinderances? It is the understanding that nothing existing in the world can provide long-lasting happiness in the long term, and that there is hidden suffering and that suffering can be eliminated] and that one is really helpless in this rebirth process if this Path is not pursued.
There is the encounter between the Buddha and Mahānāma (Majjhima Nikāya Sutta 14,) a Once-Returner and the householder cousin of Anuruddha, in which Mahānāma asks the Buddha why he can't bring himself to go forth. The Buddha tells Mahānāma that it's because he hasn't sufficiently experienced the pleasures of meditation. One can only assume from this that Mahānāma is somewhat stuck in the five hindrances (nīvaranas).
At Stream Entrant stage one comprehends the anicca nature of this world to some extent. For the mind to grasp that concept, it must be free of the “coarse defilements” or “panca-nivarana” or “five hindrances” that cover one’s mind. For that one has to live a moral life, start contemplating Buddha Dhamma, and experience the “cooling down” that results.
A Stream Entrant is incapable of actions of craving and hatred that will definitively lead to rebirth in the four states of misery (Hell, Asura, Hungry Ghost and Animal realms), but the remaining gross craving and hatred is only eradicated when one becomes a Once-Returner; subtle craving and hatred are eliminated for the Non-Returner.
The Stream Entrant is incapable of False Speech, but is capable of both Malicious and Harsh Speech, which are only eradicated when one becomes a Non-Returner. In this way, we are able to see the emergence of reasonable criteria for the Once-Returner: the emotions would be gentler, more skilful, and one would no longer be subject to any outbursts or compulsions associated with craving or hatred. The Once-Returner has basically 'broken the back' of these fetters, so that they are rendered weak and feeble.
The Once-Returner is no longer dominated by hatred or craving, that there are no longer any times in their lives when they are overwhelmed by such unskillful emotions. These emotions may arise occasionally in a weakened form, and only during awareness – they don't dominate.
In addition to having broken the first three fetters and possessing an unshakeable faith in the Three Jewels, the Non-Returner has attained complete ethical purity. It's not that they don't conceal their unskillfulness, they just don't have any, whether of body, speech or mind. They are almost flawlessly motivated by generosity, love and wisdom.
The last five fetters remain of course, but these are exceptionally subtle in comparison to what has been broken in the fourth and fifth fetters. Also, because Non-Returners are not subject to any hindrances, the Eight Jhānas are readily accessible to them.
There’s a simile where a man is suffering from pain. He gets tied up in anxiety and misery around the pain. And the Buddha says it’s like being shot with an arrow and then shooting yourself with a second arrow. The physical pain is the first arrow; the mental pain is the second one. And it’s the mental one that’s important. As the Buddha says, the enlightened person, the awakened person, may still get shot with those first arrows but doesn’t shoot him or herself with the second. It is because an Arahant has gone beyond the last of the four kinds of clingings – the Attavada Upadana (attachment or upadana is “firmly grasping”.) Attavada is the last of the attachments which is in other words a hindrance to the realization of Nibbana. So this tells us how hinderances are eradicated in stages.