There’s a long list of the most obvious changes. First there’s the darker material filmed in a darker setting and a darker tone. Then there’s the growing up of the junior actors, both physically and professionally. It’s nice to see them less stiff after 2 movies. This make the movie flow with a less contrived feeling than before.
Then there’s the focus on Harry’s character, which cuts off screen time for most of the other characters. Magically strange though, it’s the past that drives the future, yet we get to know almost nothing new of it. Harry seems slightly more mature, and magic comes more easy to him, even to a point where you finally get the feeling he’s more special than the others (where the previous movie had the other characters do more magic than Harry)
The changed actor (unfortunately, to replace the late Richard Harris) for the Dumbledore character might have been the hardest thing to swallow. With a whole other demeanor and style (which might be done intentionally ?), the change is too big to take it all at once. You will get used to it though.
Then there’s a short list of things that are the same. Character-wise, you learn almost nothing new. The story layout is mostly the same, but it’s still as enjoyable as before, even with less jokes to laugh at.8-.