Bartholomey Diaz's expedition sunk

Diaz's expedition found the Cape of Good Hope, the road around Africa to the Indies, in 1488. The event marked largely the first great leap in the era of colonialism and European domination of the world. Portugal grew from a tiny coastal nation to a great power overnight. This success is probably what truly prompted Spain to follow suit by funding the insane adventure of a crazy Genoan.

In this TL, Diaz's expedition is caught by a terrible storm along the coast of Kongo. Without Diaz' stunning success, a Portugal who's situation is increasingly worsening thanks to military misfortunes and the union of Aragon and Castille is forced to renounce largely to its foreign expeditions.

Without the stunning success of the tiny coastal state, the new Crown of Spain does not see fit to waste valuable ressources on the project to discover the new world.

Chritopher Columbus grits his teeth and head north. Paris. London. Darned Amsterdam if it must, he would find a sponsor somewhere.

Meanwhile, the lack of sea-contact with the rest of the world rapidly changes Europe. Luxury goods continue to travel through the rising Ottoman Empire at ridiculous taxation rates, ensuring the wealth of the levant and the poverty of the west. Without Portugeuse meddling, the affairs of the Indian ocean changes rapidly. Kingdoms and ports which collapsed almost overnight thanks to the hostility of mighty Portugal in OTL continues to thrive. Rapidly, the political map of the world is affected.

In April, 1497, Christopher Columbus set sail from Poitoux, with the blessings of the Crown of France and the Guilds. With a sound victory in the Hundred Years war, France is on the verge of becoming the major power of Europe and hopes to enhance her fortune by finding a shortcut to India.

Only one problem. The favorable currents that pushed Columbus westwards from Spain do not blow that far north. Columbus discovered them by chance more than by calculations.

His four ships are never heard of again. Tales comes from the north that a handful of survivors have reached Iceland, and speak grimly of the edge of the end of his armada, at the Edge of the World.

What would happen to Europe after that? How deeply would the world change?

I doubt that an event as simple as the failure of Diaz would put an end to the age of exploration. Portugal had been going down the African coast since the first half of the 15th century, and also lost quite a few ships. Unless you can provide facts supporting that the king planned to stop the expeditions at that time, I have strong doubts with the POD.

Furthermore, I am not sure that Spain would refuse Columbus in that situation, and what about John Cabot and the French expeditions?

Columbus wouldn't have failed due to a lack of currents he just would have reached more northern parts of America rather then the Carribean- parts where there is very little of interest about.
I think this has been done before, I remember a TL where Henry VII agrees to fund Columbus.

The currents in the eastern Atlantic were already pretty well known by the time of Dias and Columbus. The Portuguese in the southern hemisphere and the Spanish in the western hemisphere were able to at least partly predict which way the winds would blow in unknown territories by extrapolating from how they blew in waters that they were familiar with. No matter what nation Columbus sailed for, he would have known that the most favorable winds for sailing west in a low northern latitude were found around the Canary islands. It was already well-known that trying to sail straight west from France wouldn't work since the prevailing winds at that latitude blew towards the east.

If Dias and Columbus both fail, it could delay European discovery of the water route to the Indian Ocean and the Americas, but my guess is that someone would find both - 20 years delay at the very most. Of course, that delay along could seriously effect some of the events at the time.