There are some issues that undermine the fascist label when it is applied to Franco and the government he set up. He professed devotion to traditional power structures, specifically the military, the Catholic Church, and to monarchism. That his devotion to these principles was often superficial is also evident through his actions, but he did not seek to dismantle all of the old in order to invent his new state. This was a different approach from Germany or Italy, for example.

The fascist party was indeed part of Franco's one party system, but they got less than 0.01% of the votes in the 1936 Spanish General Elections (less than 7,000 votes total). They had power out of proportion to their numbers via their high gun ownership rates, but their leader (José Antonio Primo de Rivera) was imprisoned before the war began and was later executed by the Republicans, which allowed Franco to subsume the Falange into his own single party system. The fascists were only part of the larger party.

None of this is really exculpatory for Franco's regime. The best word I have seen for it is "barbarous" from Jose M. Sanchez's The Spanish Civil War as a Religious Tragedy. Paul Preston's The Spanish Holocaust gives a detailed account of Nationalist violence against noncombatants both during and shortly after the war, and it is a chilling account. Preston estimates that there were 200,000 killed by the Nationalists outside of combat--a staggering number. That Franco was more interested in advancing Franco than fascism doesn't excuse those crimes. If I wanted to sound academic, I would call Franco a reactionary authoritarian. If I wanted to quote /u/Georgy_K_Zhukov, I would call Franco a Francoist. I wouldn't call him a fascist (nor would I apply that label to his government), but that doesn't mean much in my book.

Stanley Payne is the foremost expert on Spanish fascism, but I haven't read most of his specialist work on that subject as my interests have laid elsewhere. Preston and Thomas spend a couple pages on this subject in their one-volume histories, but they don't dwell on the distinctions. They both cover Franco's process of placing himself at the front of each of the factions that made up the Nationalists in good detail, though.