Ultimately, very little: their leafletting campaign was very localised, and clamped down upon very quickly.

The nazi regime was not brought down by internal pressure, it fell because an invading army conquered the regime's capital concurrently with invading armies conquering other parts of their territory. The regime fell because it was physically unable to exert control over the population anymore.

Sophie Scholl was executed in February 1943. Whilst the war had turned against Germany at this point, the blunting of Germany's attacking forces at Kursk in Summer 1943 and then the catastrophic defeats of Operation Bagration and the western allies' invasions of France in Summer 1944 had yet to happen.

Morale amongst the German population had been falling long before the end, and the regime's promises of total victory seemed hollow, but there was stout resistance from Nazi forces right until the end. German soldiers were still visiting their homes on leave, and telling their families of the succession of defeats that was bringing the enemy closer to the heartland, and it was this news that weakened the people's resolve far more than an isolated leafletting campaign.

Where the White Rose has become important is in post-war German culture, as an example of moral courage in the face of overwhelming odds - a set of heroes who "did the right thing" even though it could have only a small impact and little chance of changing anything, but was still the right thing.