These are the three best-known British folk heroes, and each has a different archetypal quality.
Arthur is the earliest - dating from the time after the Roman Legions had left the island, and associated with a decades-long period of successful British resistance to the Saxons. He is the archetypal Good King, Father of the nation (although, strangely, not an actual father of children - except Mordred...) who presided over a golden age; brought-down by human frailty and supernatural evil.
Interestingly, the documented historical figure that most resembles Arthur is Alfred the Great - warrior, scholar, lawyer and devout Christian; who was, of course, a descendant of the Saxon invaders whom Arthur resisted. This perhaps shows that Arthur is an archetype of land and spirit, not primarily a matter of genetic descent.
When modern people think about Arthur there is the yearning, and
perhaps hope, for Arthur to return (revive, reincarnate or as a modern
spiritual descendant) and vanquish the powers of evil and reinstate the
Merlin is from a generation or two later; and is perhaps a composite of a Welsh poet and wizard with a Scottish-English border seer and prophet. Despite his well-attested decline from high rank into exile, madness and poverty - and a well-known 'grave' near Peebles in Scotland; the most imaginatively vivid legends have Merlin's time being curtailed by spiritual imprisonment (in a crystal cave, or oak) - with the promise of eventual return.
Spiritually some regard Merlin as the last of the (good) druids; others as a transitional figure - with one foot in pagan Druidism and the other in the new Christianity (of the Celtic variety).
Modern English people have assimilated Merlin in many archetypal versions, from Gandalf, through Doctor Who, to Dumbledore. We see him mainly as a potent combination of magician, prophet, seer and wise-man; but in an eccentric, unpredictable, 'irresponsible' personality. Someone who operates behind the scenes, indifferent to power, status, wealth - probably also indifferent to sex, marriage, children and the like.
This latter idea finds an allegorical equivalent in the stories of Merlin's conception as a devil and a nun with a monkish baptism; or a virgin and an incubus; Merlin therefore having some aspects attributed to Jesus and 'the light', but mixed with something darker and more instinctive.
Robin Hood is 'the people's hero' from the Middle Ages, leading a successful resistance to the Norman aristocracy. Robin is not much of a spiritual figure, but exponent of a 'pastoral idyll', a paradisal life of leisure, music and poetry - hunting, competing and helping the needy; lived-out in beautiful English woodland.
Robin is the escape from authority, liberation from work, brotherhood of all Men - and Robin's men are a fellowship of oddballs, eccentrics and drop-outs. Again, he is not a Father. (We Britons seem uninterested in patriarchs when it comes to folk heroes!)
Interestingly, Robin seems never to die, but to be a permanent presence - always young, always the same; always in his role of the impulsive and pleasure-seeking but honest and kindly counter-cultural rebel. All this makes him seem kin to the fairies, or a nature spirit - although Robin Wood is not magic except for his supernatural skill with the bow.
Of these folk heroes of Albion, the one who most appeals to me, and who seems to be most what we need, is Merlin. This is because I think we cannot - and should not - look to being rescued by a King, nor saved by an outlaw. We need to help-ourselves; and what we therefore most need is the wisdom of a wizard.