There was one key distinction between the two: the Spanish inquisition managed to get itself under the direct power of the king of Spain, where the Roman inquisition depended on the Pope and the Vatican. I'll explain.
As a rule of thumb, inquisitors and other religious officials dealing with cases that were thought to require "special investigations" (such as the charges of homicide against the French nobleman Gilles de Rais, who was convicted and executed for the murder of roughly 140 children) needed to contact Rome and wait for its approval of an interrogation request using torture and any sort of verdict which had to be ratified and confirmed by the Holy Office. This was in fact put into being in 1542 in a broader effort by the Church to control more efficiently its many appendages. This meant that inquisitors didn't work anymore under the jurisdiction of their bishops and didn't have the same freedom of leverage, also because it was the public, secular authorities whom executed the tribunal's rulings.

Spanish inquisition functioned much in the same way, but it bypassed the need to have all its decisions controlled and approved by the Pope; the king would do so instead. The most relevant theory for this "special need" of legal freedom was due to Renaissance and Modern Spain having huge populations of freshly converted Jews and Muslims, most of whom didn't convert willingly and thus had great resentments against the Christian majority. The mocking term used for them, marranos, was meant to bring infamy and shame of not being "old Christians". Since it seems of having been a sizeable portion of converts secretly practicing their ancestral religion, people began denouncing them to the authorities (one of the most common charges seems to have been the lack in pork meat consumption).

Sources:

Levack, Brian P., "The Witch-Hunt in Early Modern Europe". London: Longman, 1987;
Romeo, C. "Inquisitori, esorcisti e streghe nell'Italia della Controriforma", Sansoni, 2003;
Romeo, C. "L'inquisizione nell'Italia moderna", Laterza, Bari, 2009