What is important to understand about the Franco regime is that the coalition that formed the government was largely ambivalent - while the fascists around the Falange led the country, their political power was mostly supported by religious groups, monarchists, business owners, conservative factions and military alike. That means that unlike Italy, Spain's government wasn't as structured or as straight in its decisory processess, and in fact had to content and politically manage a plethora of different groups that served as base for the government. Professor Wayne H. Bowen from the Department of History of Southeast Missouri State narrates a sentiment of doubt and uncertainty after the Spanish Civil War - would Spain join the Axis Powers and declare war alongside Italy and Germany? The Luftwaffe operation was certainly a factor for the Nationalist win during the Civil War, and combined with ideological overlap and public support is safe to assume that Franco had some partiality towards Hitler and Mussolini; that being said, he still would have to maintain the support of the right-wing groups that formed the basis of the government, which meant that the decision to support or not the Axis wasn't trivial. Bowen compares the Francoist dictatorship with Romania, Hungary or even Portugal, in which while the fascist party was certainly a political agent of importance, it was ultimately subordinate to the traditionally conservative state authority and military. Not to mention the struggle to deal with opposition factions or the naturally diverse cultural landscape of Spain, which meant having to manage the Basque and Catalan movements. All that picture is meant to show that while ideologically Franco definitely had some inclination to join the war, politically or pragmatically that were also a multitude of incentives to not necessarily do so. (Spain did eventually send a famous division to the Eastern front, the famous División Azul, but it wasn't a full-fledged, nationwide contribution to the war effort). Paul Preston, another Hispanist specialized in the twentieth century, puts it better than me:
"As an openly declared enemy of liberal democracy and Bolshevism Franco could not conceal his sympathies when Hitler unleashed his war to exterminate both. In the last resort, however, the Caudillo’s natural inclinations in foreign policy were restrained by two overriding considerations: his own domestic survival and Spain’s economic and military capacity for war."
That being said, it's worth mentioning the participation Franco's minister of interior, Ramón Serrano Suñer, in developing close relations with Italy. He certainly wasn't the only Fascist enthusiast on the government (or even less the only one discontent and distrustful of liberal democracy) but he was a major component in the Spanish administration during the early years of the war and definitely contributed to the general feeling that the Spanish participation in the conflict was imminent. When war did broke out however (specially after Italy joined the war in 1940), even Suñer was skeptical about the capability of the state in participating. It became even more blurred after it became clear that Germany wasn't going to achieve a swift victory. In the end, while there definitely was enthusiasm towards the Axis in the Spanish government, it was never quite pragmatic or quite rational for Spain to declare war, which earned quite a bit of a favour with some of the major powers that later would be part of NATO (specially Britain, whose political and strategical history with the territory of Gibraltar is notorious). We didn't even touch the matters of colonialism and the interest in Morroco and other African territories, but that is going to be a matter for another time.
EDIT: I'm making this edit to clarify a bit some matters about foreign relations post-war and the general policy shift from Francoist officials during the last years of the war. First of all I want to encourage anyone interested in reading about Franco, the Civil War and really about Republican Spain in general to go after Preston since he's probably the great specialist writing in English about this stuff. Anyway, the point I want to make is that became clear the the American contribution to the war effort, specially on the industrial side, was going to be instrumental and there was very little chance for Germany to turn the tide of war back again, Franco began to focus on his common disdain for communism rather than his distrust of liberal democracy. Again, on the words of Preston:
" On 18 October 1944 he (Franco) wrote a letter to the Duke of Alba, the contents of which he was asked to pass on to Churchill. In it, he proposed a future Anglo-Spanish anti-Bolshevik alliance. In Franco’s Hitlerian analysis, ‘after the terrific test Europe has gone through, those who have shown themselves strong and virile among the nations great in population and resources are England, Spain and Germany’. However, Germany, along with France and Italy were all incapable now of standing up to Russia. American domination of Europe would be disastrous. Accordingly, Britain and Spain together should work to destroy communism. He dismissed his own pro-Axis activities as ‘a series of small incidents’. The only obstacle, he claimed, in an astounding act of dogged myopia, to better Anglo-Spanish relations during the previous years had been British interference in Spain’s internal affairs, in particular the activities of the British Secret Service."
Basically, the whole gist of the thing is that Franco tried to shift his international policies to focus on what had clearly become the new problem for the rest of the Western powers, the Soviet Union. This strategy was met with varying success on some fronts, but it was the explanation on why Spain became much more collaborative with NATO countries during the Cold War.
Bibliography
Preston, Paul. (1990). The Politics of Revenge: Fascism and the military in twentieth-century Spain.
Bowen, Wayne H. (2005). Spain During World War II.