A key element missing musically from the post-Williams Potter films (aside, of course, from Williams himself) were scores that didn't just support the films they accompanied but that also dared to interact with and engage in helping to create the visual narrative as well.
I think a great example of this is in the first film when Harry first discovers the Mirror of Erised and looks into it to see his dead parents standing alongside him. The scene shows Harry's entire approach towards the mirror and his revelation. Now, Williams could have easily, and quite logically, chosen to start playing his wonderful family theme for the film right when we first make the discovery alongside Harry of what the mirror can do (i.e. when we first actually see his family - at 1:04) and it would have fit nicely and hit the proper emotional beats and what have you. Instead, Williams holds off, instead scoring Harry's approach and discovery with something a little more somber but still effective. It isn't until Harry himself reaches out and touches the mirror (1:40), physically both reaching out with longing for something he can't have and seemingly summoning the next bit of music in the process, that Williams allows the family theme to be heard.
It's so subtle but so incredibly genius. You don't even really notice it if you're not paying attention to it. On one hand, you have the visuals dictating the score in that it isn't until we see Harry touch the mirror that we hear the theme. On the other, we also have the score interacting with the visuals and, by extension, dictating our own emotional experience because by delaying the usage of the theme Williams is picking and choosing when we, the viewer, get to feel that maximum emotional impact. On an even larger level, Williams doesn't use the theme until Harry himself, by virtue of being so overcome with longing that he is moved to physically react to what he is seeing, reaches his most emotional moment of the scene, and, by waiting until then to play the theme, so also allows us to share with Harry in that emotional moment. Just sublime.