I understand that creatures like vampires and the Fair Folk are "slaves to their nature", but I think wizards and practitioners aren't comparable if they don't use black magic or enter into demonic pacts and the like. So maybe I could get a bit of clarification?
Mechanically, it works like this:
Characters have a pool of points, called fate points. Each session it begins filled to a certain point. This point is called your refresh. So if your refresh is 5, your fate point total begins at 5 each session. Abilities beyond the norm are purchased as Stunts. Stunts reduce your refresh. Things like magical power, super speed, and the like come as Stunts.
Here comes on the scene Aspects. These are things about your character that are both narratively and mechanically important to the game, with both an up side and a down side. So an Aspect, "Always polite to women," might be possible. You can use a fate point with such an Aspect to add to a roll. Let's say you're convincing a woman to let you past security. You spend a fate point and justify the bonus you get by saying that she's impressed by how polite you are and thinks you are trustworthy.
Remember the bit about a bad part? Well, those are called compels. The person running the game (or even you) can force you to act differently by compelling an Aspect. These complicate your life. You might be about to do something and the person running the game might point out that doing that would be really rude to the woman you're talking to. You can then either accept a fate point and act accordingly or spend a fate point and do whatever you want.
This is where freedom comes in. The higher your refresh, the more fate points you're likely to have. The more fate points you have, the more compels you can resist. The more compels you can resist, the more you can do whatever you'd like.
Since Stunts -- from which magical power is derived -- reduce your refresh, they cause you to need to give in to more compels. This reduces your freedom of choice somewhat, therefore magic stuff reduces your freedom of choice.
Remember all those times where Harry can't use technology or the like? Those are compels against some applicable Aspect in game terms. You know how he is chivalrous to a fault? Likewise, a compel. In the thinking of FATE, that is a decrease in free will.
Before you get worried that you're going to be forced to do things you don't want to, that isn't how things work. You choose your own Aspects. You choose exactly how successful compels affect your behaviour. By the wording of an Aspect and how you choose to react to it, you're choosing what problems you'd like your character to face. Characters not facing problems is boring.
In terms of the books, it's not exactly the same. Harry doesn't lack free will because he can use magic. He is, however, pushed into a variety of situations because he's a wizard. He lives in "interesting times," as it were. Normal folks around him lack quite that weirdness magnet thing going on in his life.
So it's not that wizards lack all free will. It's that wizards live complicated lives. The more powerful they are, the more complicated their lives get. Real ass kickers are ruled by their passions, chased by their pasts, and bound to make dreadful decisions.