I have a fourth character to add to the jumble of Mr. Rochester from Jane Eyre, Henry Higgins from Pygmalion, and Heathcliff from Wuthering Heights. The newbie is the title character of The Phantom of the Opera. I mean him no offense by calling him "the phantom" instead of his name, Eric; it's just that most people don't know his name. Even less people than know that Sleeping Beauty's name is Aurora, I'd say.
The most obvious similarity here is the controlling characteristic. Out of the four, the phantom seems the one who has the most control. The entire opera is under his control. Heathcliff only controls a couple of houses out in the country, Higgins only controls his work and everything relating to it, and Rochester acts like he's in control, but really isn't. He has a past and present with Bertha that he can't escape and Jane chooses to leave him.
With the exception of self-centered Higgins, they're all exerting this control, to a degree, for a woman. The phantom sets up his elaborate scene to gain Christine, like Heathcliff wants revenge for Cathy and Rochester wants to be able to have a future with Jane.
The phantom is very similar in my eyes to Heathcliff with the buried-deep good core beneath so much evil from the pain of the past. (That's a nice, wordy description, isn't it?) He, like Heathcliff, couldn't be accepted by the world, because of his appearance versus his lack of social standing. He used the excuse to cause torment to others. Christine isn't much like Cathy, though, except that they both choose another man. Christine is a better person, which is why she doesn't love the phantom back, even if she can and does pity him.