Not much would have changed, according to Mark Grimsley. The Spanish would have seized Gibraltar with German assistance, which would have complicated the Allied war effort in the Mediterranean, but the impact on other fronts would have been limited. Post-civil war Spain simply lacked the resources and the political cohesion to make a critical difference in the war. Franco, who consistently resisted entreaties from Hitler and Mussolini, seems to have largely agreed with this analysis.
In Advanced Third Reich and its successor, World at War, Spanish entry can help put a nail in the coffin of a weak Britain in 1940 or 1941. Failing that it becomes a liability; Spanish control of Gibraltar tends to dissuade the Allies from wasting resources in the Mediterranean (this effect would likely have happened in the real world, as well), and the Axis simply lacks sufficient forces to defend the entire Atlantic seaboard from Allied amphibious assault. Since the game substantially understates Allied air and naval dominance from 1943-1945, I suspect that the overall net effect of Spanish entry would have been an extension of the war in Europe by a few months, and the end of Franco’s regime.