Short answer, Franco was out for Franco and Spain. He didn't trust Hitler and wasn't out for some super-national fascist alliance. He was open to the idea. And he negotiated with Hitler and Mussolini throughout the war. But he wasn't going to join the war, unless he was sure Spain was going to come out a winner, and with significant gains.

At various points he was asking for parts of Southern France, North Africa, Gibraltar, major construction to fortify the Balearics, and huge supplies of grain and oil to make up for the imports he would lose from the West. Of course this was well beyond what Hitler was interested in offering.

Instead Spain used the war to send away the radicals. The fascists volunteered for a Spanish division that served in Russia, the Blue Division. And those opposed to Franco frequently found themselves serving in Allied armies (usually the Free French). There was a new resistance in Spain (the Maquis). But Franco was able to stamp it out.

And Franco traded with both the Allies and the Axis throughout the war, favoring the Axis generally until 1944 or so.

By playing both sides Franco was disliked by both sides. But not enough to get rid of him. Stalin wanted a declaration of war. He wasn't a fan of the 45,000 men who served in the Blue Division. But Churchill talked him down into some sanctions and keeping Spain out the the UN for a decade.

Stan Payne came out with a book not too long ago covering just this topic "Franco and Hitler" it is well worth reading.