Well, I do think Henry VIII's over-sized reputation is largely in the Anglo-American tradition where it is typical to focus on English monarchs. For instance, Richard I and King John tend to get more attention than their sometime ally and ultimate rival Phillip II of France, although Phillip was one of the most important kings in French history.

And it would be naive to say that Henry's personal life didn't play any role in his the fascination with him. I would say this is a small part of why the Tudors captivate, as a whole. And the Tudors have an undeniably great narrative story from beginning to end. For Henry it's also that his life story has a fable like quality. He goes to great lengths to have a son and both his daughters end up ruling anyway. It's the "careful what you wish for" narrative that has long been popular.

But, all of that aside, yes Henry was important. First and foremost, Henry broke with Rome. Now Henry would claim he remained a true Catholic and he never embraced Protestantism. But breaking with Rome had a huge impact on the growth of England and in the narrative of the Reformation. England was the the most important area to do so at that point in time and it seemed to pave the way for others to follow.

Although Henry was never a protestant the break with Rome did give Protestants in England some hope and ability to come out of hiding (although still to their own risk.) Many in his closest circle, including several of his wives and their families, were Protestant after the break. The religious settlement that came under his son, Edward, and then revived under Elizabeth (Elizabeth's religious settlement was largely Edward's in doctrine although she gave more leeway in some of the traditional superficial aspects of the old church) would not have happened the way it did had Henry not broken with Rome. Henry did not create the Anglican church we know of today that was largely Edward and Elizabeth with some later reforms. But he absolutely made it possible. And that, in and of itself, is enough to be a very important figure in history.

The break also had deep societal implications beyond just the change in worship. The Church had been responsible for housing and feeding the poor and sick. The Church was also an unparalleled fundraiser so it encouraged others to give to this purpose as well. With the breakup of the monasteries and the traditional Church affiliations there was a real gap. Under Henry VIII the poor laws began to be passed. These were quite harsh by our standards and punished able bodied people for begging (without much understanding of economic factors). But later poor laws (continuing under Edward and Mary) also mandated that local parishes (which are now governed by the crown) had to give to the poor. The Elizabethan poor laws mandated how the poor should be registered, collections for the poor, and required that local parishes put the poor to work.

This matters because it's when the government begins to truly involve itself in legislating these sorts of domestic matters. And the idea that the monarch really must involve himself or herself with concerns like poverty (even if it is just to punish it.) These were not crown functions in the Middle Ages.

Another important change under Henry VIII was the way he used Parliament for legislation. Now, of course, this did not begin under Henry VIII. But Henry really used Parliament in new ways especially through Thomas Cromwell. It was Parliament that issued his declarations on religion. He used Parliament to determine the succession and to declare his daughters bastards. Parliament was making legislation in expanded ways.

Unlike most of his successors Henry really had Parliament doing his bidding to a large extent. And he set some precedents that his successors would regret. But, in retrospect, it was really a turning point in the scope of Parliament.

Henry would have wanted to be remembered as a great warrior king. He would have loved to conquer France. He wasn't that. But he did break with Rome which had radical implications for not just England but all of Great Britain and, arguably, much of Europe. Whether or not he was a good king is something that could be debated to the end of time (I would say that both Henry VII and Elizabeth I were the better monarchs but that's my opinion, which is worth nothing.) But I don't think his importance can be denied.

ETA: I meant to say England was the most important area to break from Rome at that time. I didn't mean to deny the importance of the later great Protestant states.