Cape Verde was not discovered a couple of years before Columbus first voyage, but 30 years, as the Portuguese had founded a city in 1462 by the name of Ribeira Grande, nowadays called Cidade Velha. This outpost was of paramount importance for the Portuguese, as it was a link with the Coast of Slaves, and later with Elmina (today in Ghana), the most important trade spot for slaves, and other resources like gold. Such was its importance that you can see Elmina (La Mina or Sao Jorge da Mina) depicted very richly in Portuguese maps from the very late XV century.

But let's not get ahead of ourselves, as we have to talk about Cape Verde and how come the Spanish did not use it as a starting route, opting instead for the Canary Islands.

In 1474 King Henry IV of Castile died, and the legitimacy of his daughter was more than dubious. So dubious it was, that Henry had recognised his sister Isabel as the heir to the throne. In this cases, it happened what ought to happen: yet another civil war in Castile. On one side there was Joanna, allege daughter of Henry IV, backed by her husband the King of Portugal. On the other, Isabel, backed by her husband Fernando, King of Sicily and heir to the throne of Aragon. This Castilian civil war ended in 1479, and as it happens with all wars, it ended with a treaty, the Treaty of Alcaçovas. Amongst the many things contained in the treaty, there was a provision on the Atlantic navigation. I quote:

They shall not disturb, bother, or discomfort, neither de facto nor de iure, consciously or unconsciously, those lords King and Prince of Portugal, nor any Kings of Portugal for the time being, or their kingdoms, the possession or quasi-possession that they had, on any deals, lands, or commercial agreements of Guinea, with its gold mines or any islands whatsoever discovered or to be discovered, isles of Madeira, Porto Santo, Deserto, and all the islands of Azores, and the islands of Flores near Cape Verde, and all the lands that are now discovered, and any whatsoever islands to be discovered or conquered, from the Islands of Canary southwards against Guinea, because everything that has been discovered or may be discovered, conquered or discovered shall be from those terms onwards be for the Kings and Princes of Portugal and their kingdoms, excepting the islands of Canary, those being Lanzarote, La Palma, Fuerteventura, La Gomera, el Hierro, la Graciosa, la Gran Canaria, Tenerife and all the other islands of Canary won or to be won, that are left for the kingdoms of Castile and León.

The treaty is abundantly clear on who did these islands belong to, so Columbus, or for that matter any other seafarer affiliated with the Crown of Castile, could not use that as a base of operations, and that is the reason why Columbus went from Palos (nowadays Palos de la Frontera) to the Canary Islands, and form there he journeyed to the Indies.

This division changed with the Treaty of Tordesillas in 1494, where the line of demarcation went from being a parallel to being a meridian. This treaty was absolutely necessary, as the lands discovered by Columbus were not under the scope of the Alcaçovas treaty. They were indeed south of the Canary Islands parallel, but they were not "against Guinea".

Sources:

Fernández de Navarrete, Martín (1825), «Primer viaje de Colón». Colección de los Viajes y descubrimientos que hicieron por mar los españoles desde fines del siglo XVI. Madrid: Imprenta Real

Álvarez Palenzuela, Vicente (2006), «La guerra civil castellana y el enfrentamiento con Portugal (1475-1479)». Alicante: Biblioteca Virtual Miguel de Cervantes.

Ramos, Demetrio (1995), Colón en Simancas. Valladolid: Sociedad V Centenario del Tratado de Tordesillas.

Tratado de Tordesillas (1973), facsimile from the original Spanish and Portuguese documents. Madrid: Ministerio de Educación y Ciencia.