There are several reasons for this. Firstly, while yes both Italy and Germany supported Franco during the Civil War, the Civil War ended in April of 1939, and essentially left the Spanish population displaced and in ruins. World War II began five months after the official end of the Civil War, however with an economy in ruins, Franco of course focused attention to internal priority, which included massive political repression. That being said, Franco did support Germany and Italy in terms of ideological similarity, but did not wish to fight against the Western democracies because he did not share the same antipathy as the others felt, having kept out of WWI and avoiding the economic punishments which Germany predominantly absorbed. Franco did allow Spanish citizens to volunteer and fight alongside the facsists against the Soviets in what is known as the Blue Division, however because he did not share Hitler's anger towards the west, he did not allow Spanish volunteers to fight against Western democracies.

Also, while Hitler desperately wished for Franco's support, the two parties could not actually agree on key issues. Franco wanted the guarantee to annex Gibraltar and parts of Africa into a Spanish Empire, however Hitler could not promise these, as they also went against the territorial expectations of Mussolini.

Spain is immensely Catholic, and since Franco was a strong Catholic he was both dismayed and distrustful of Hitler's anti-Christian rhetoric and glorification of "Nazi mysticism."

Lastly, Franco did support the Axis powers with regards to equipment and naval vessels, however when the tide of war drastically changed in 1943, Franco recognizing the Axis powers would lose the war, effectively ceased any collaboration with the Axis powers.

EDIT: TL/DR: Franco displayed real politik and didn't wish to condemn Spain to an unfavourable outcome.

Spain's non-involvement and neutrality with the Second World War has been reevaluated by a number of historians since the 1980s. Franco after 1945 presented his non-action as evidence that he was more anti-communist than fascist and he used both Hitler and Mussolini to defeat Spain's communist movement and then had the last laugh by refusing to join in the war. Called chaqueteo (turning of the coat) this is obviously a political and self-serving narrative and it obscures Spain's numerous pro-Axis actions and emphasizes Franco's Realpolitik genius. In reality, Spain maintained a form of neutrality in the war that privileged Axis interests and a number of external events and timing precluded Spanish entry into the war.

Germany initially approached Spain during the Second World War at first as a potential partner. Franco's ideological preoccupation with anti-communism prevented any move away from Spain's neutrality. Conservative elements within Franco's government, the Church and monarchist military officers in particular, looked askance at the Molotov-Ribbentrop pact and did not want to engage in a war that would see the Nationalists in an alliance with the USSR, even if an ally by proxy. The unwillingness of Hitler to treat his alliance partners on anything close to equitable levels also meant that the Germans would not offer any concrete assurances of territory or independent action. This meant that negotiations with Spain when Germany had the most to offer her, roughly the time period of the fall of France through mid-1941, were plagued by these two stumbling blocks. Therefore, the ideal window for Spanish belligerency passed.

Barbarossa seemingly removed this obstacle, but other issues propped up. The increasing effectiveness of Allied airpower and naval forces put pressure on the Spanish economy and made prospects for Spanish victory less apparent. Although Franco and some Falangist extremists like the Foreign Minister Ramón Serrano Suñer still believed in German victory, more level-headed army officers maintained skepticism. The army demanded and received a more moderate replacement to Serrano Suñer in the form of Francisco Gómez-Jordana, a military general and conservative monarchist with Anglophile tendencies. Allied bribes among army circles and threats also put a damper on the enthusiasm for the war. The creation of the Blue Division of Spanish volunteers actually helped defuse tensions and counter-intuitively bolster Spanish neutrality. The volunteer nature of the Blue Division eliminated Falangist attacks on Spanish neutrality and removed some of the most fervent anticommunists from Spanish domestic politics.

German strategy with regard to Spain shifted after Barbarossa and also played a role in limiting Span's options to enter into the war . From 1941 onwards, the Germans mainly wanted three things from Spain: volunteer soldiers, workers, and raw materials. There were very few voices within German diplomacy that wanted Spain to actually become a belligerent. Germany increasingly began to see Spain's role in the Second World War as analogous to the role played by the Netherlands during the First World War: a "window" to escape Allied blockade. Spanish exports of tungsten became an essential component of the German war effort as tungsten was an essential mineral for both high-end machine tools and anti-tank ammunition. Franco took advantage of this shortages to restore Spain's gold reserves and pay off some of its debt to Germany and Italy. The Allies also competed with Germans to buy up Spain's tungsten reserves and used other inducements such as an embargo on oil to pressure Spain to break its economic ties with the Axis. Despite this competition, Franco's focus on autarky meant the Spanish economy barely grew during the war and the wartime economy meant shortages and suffering for the Spanish people. Germany would use the friendly nature of Spain's government for its espionage networks and covertly use Spanish territory for refueling U-boats with the silent acquiescence of Franco. Spanish belligerency would eliminate these benefits that Germany was reaping by Spain remaining one of the few genuinely friendly neutrals. Although Spain did not turn its cloak, the Germans increasingly did not Franco to do so and put relatively little pressure on him to join the war after 1942.

Sources

Bowen, Wayne H. Spain During World War II. Columbia: University of Missouri Press, 2006.

Payne, Stanley G. Franco and Hitler: Spain, Germany, and World War II New Haven: Yale University Press, 2008.

Pike, David Wingeate. Franco and the Axis Stigma. Basingstoke [England]: Palgrave Macmillan, 2008.

fyi there's are a few posts on Spain in the FAQ

• Neutral countries during WWII

there are a lot of fine answers here. The only other thing I would add is that Spain was very vulnerable to being cut off from her overseas supplies of meat and grain. I think these would be from South America. If they went to war it would not be hard for the allies to put a lot of pressure on them by naval blockade.

Maybe others can elaborate on this, I am not expert in this area.