The problem with widespread corruption starts, like most others, with the power structure of the country.

When Italy unified, it was only superficially under a strong central government. In reality, the ruling dynasty and the central government that it created in Rome allied with local potentates, leaving them to administer their own regions as long as they fell in line on key things. This led to a central government that was institutionally incapable and unwilling (disinisterested, I'd say) to fight corruption in the periphery, and you'll notice that most Italian corruption by far happens in local governments, with the national government almost never suffering from significant corruption scandals (other scandals yes, but not corruption).

Contrast this with the unification of Germany, which happened around the same time. The German government directly recruited the various ruling classes of the old German states, some old and some new, to the service of the state; working for the central government was the main way to power, prestige, even wealth (in an age where industry was often tied to state affairs). That didn't erase corruption, but it made it so that the ruling classes themselves were little tolerant of it, because it weakened the state and therefore themselves.

Basically, Italy sucks balls because we were unified and then ruled for decades by one of the most spineless and despicable dynasties of all time, the House of Savoy. Germany rocks because it was unified by Bismark and dominated (initially) by a Prussian state that was determined to get everything squared away under its own successful model.

Of course all of this is speaking in broad generalizations, but I think it gets the gist of it.