The Spanish Inquisition was less about religion than politics. Contrary to propaganda (mostly English), it wasn’t terribly violent and preferred to boot people out of the country.
The key is that the now-united Spanish monarchy had been struggling for centuries with the Moors, along with periodic waves of invasion from North Africa. They decidedly wished to avoid that in the future, which is one reason they established a protectorate (read: border zone) in Morocco later on.
When they captured Granada, many Muslims “converted” but kept their old ways. Thus, they formed in the eyes of the Crown a dangerous and unreliable group which might support Islamic powers trying to invade Spain, or simply rebel themselves. Thus, the Inquisition, intended to root out, depending on your view, heretics or closet Muslims, with Jews thrown in for good measure. This sparked an exodus to Morroco and Algiers.
Was it cruel? Yes. Was it necessary? That’s a different question and much tougher. Having fought over the land for generations upon generations, the monarchs of Castille and Aragon were not about to endure any more, and took what steps they felt were needed to deal with it permanently.
Contrary to that respected historical source Monty Python. EVERYONE expected the Spanish Inquisition as each “Auto-da-fe” was always proceeded with a 40-day grace period during which anyone was free to confess (or to accuse others).
That’s one reason for the name. Another reason is that people, specially foreigners, don’t realize there were several Inquisitions within Spain, and locations which never had an Inquisition (a religious tribunal dedicated to religious “crimes”) but where charges of witchcraft or heresy were investigated by civilian tribunals; the one that’s famous abroad albeit with the wrong name, and which has actually been referenced several times in this same thread (dates given, etc.) is the Castillian Inquisition.
There were several separate Inquisitions for Castilla and for some of the territories of the Crown of Aragon. Other territories within Aragon had no Inquisition; Navarra had one on and off (more off than on; for example, the Salem-style witchhunts of Zugarramurdi took place under a civilian tribunal).