Definitely separate kingdoms. Both Kingdoms were very difficult, and full unification would have been very difficult to achieve, and it is debated over whether the monarchs sought it.

On the one hand, there were some efforts to attain equality between the crowns. In Castile, royal documents had the crests of both monarchs, and Isabella granted Ferdinand power to rule in her absence. The three principle gold coins in Castile and Aragon were given equality, and some customs duties between the two kingdoms were eased.

However, these were only minor reforms. Customs duties, although eased, still remained between the two kingdoms. Other than the Inquisition, there were no royal institutions that worked in both Aragon and Castile. There were also no attempts made to unify the laws and legal systems in the two countries.

To see why this was, you have to look at how different the two kingdoms were. Castile was formed through the Reconquest campaigns, meaning it hadn't developed a feudal system which most other kingdoms had, so they had much more powerful nobles, and less royal authority. Aragon (or the separate realms within Aragon: Aragon, Catalonia and Valencia, as well as Mallorca and Sardinia in the Mediterranean) were much older, and therefore had developed feudalism, so there was a lot more stress on the rights of the king and his subjects, their freedoms (fueros). This was reflected in the laws of the two kingdoms, meaning they were very different, and almost impossible to combine without some conflict. Aragon was also much smaller than Castile, meaning they would inevitably become overshadowed.

So, while the two kingdoms did work together, especially in foreign policy, you could not say that at this point they were unified into a single kingdom.

To add one tiny note to /u/potpan0's answer, the exclusive use of "Spain" is a relatively modern construction. It (España) was the name used in the constitution of 1876. Prior to that, it was largely called "the Spains" (las Españas) as well as "Spain" pretty much interchangeably. For instance, the 1812 Constitution used "las Españas" rather than the singular form. The tension between regionalism and nationalism in Spain has a long and complicated history that tends to get glossed over in summaries to avoid confusion.

A little outside the scope of your question, but this division had some interesting consequences. Columbus' voyages were funded by Castile, and most of gold from the Americas went to Castile, not Aragon. Then two generations later when Charles V was burning Spanish money in a failed attempt to keep The Netherlands from becoming Protestant it was actually largely from Castile that he was getting his money. This was because Aragon had more rights to refuse their king than Castile had.