While I enjoyed The Cutting Edge, I do not think that it is in any way comparable to Mimi. Though the characters in The Cutting Edge were interesting, they had little internal depth. I may be harsh, but they did not know what they were doing, did not know what they should do, and did not know what they wanted. Both leads were arrogant, irrational, and unwise (kashikokunai) characters who acted in whatever way they felt like and had little consideration for the effects, especially long-term ones, of their actions. I enjoyed the film, but I could not take such characters seriously. The Cutting Edge is a movie I would like to take my sweetheart to, but unlike Mimi O Sumaseba, I would not bother watching it alone or with other kinds of friends.
In contrast, the characters in Mimi were very much alive. The story was more than simply having the characters figuring out which person of the opposite sex they wanted; the characters were living for something more than just themselves, namely artistic devotion. (It is such artistic devotion that gives us all the countless historical works of art in all fields.) Seiji knew that he had some talent in violin-making, and he endeavored to pursue his dreams. Shizuku got to know him as a great person (subarashii hito), a young man who had a goal in life, not just as "a cool guy" (kakkouii hito). She began to wonder if she would be able to do something like that too. The story is not just about a girl wanting a guy to be her boyfriend or husband; it is about two young people trying to grow up to become great people.
Some people may find the final proposal scene "over-dramatized" and hard to believe, but it was perfectly believable to me. To tell the truth, I was surprised when there was a burst of laughter at that scene during the CJAS showing because I personally found the scene very touching and not "funny" at all. Considering Seiji's seriousness (majime) at the things he was doing, and the maturity of his thoughts and actions (compared to the average 14-year-old kid who is just hanging around), the proposal was very natural and not awkward at all. The qualifier, "ima ja nai kedo..." ("although not now..."), reinforced that Seiji knew what he was talking about. The seiyuus' acting also contributed a lot to the validity of the scene: both the words used and the way that they were said were a major factor in making it one of the best ending scenes in movies (anime or otherwise) I have ever seen. (I could imagine the scene losing its charm and actually sounding funny when dubbed...)
I'll compare my opinions on Mimi with those other CJAS showings that David cited. I agree that Video Girl Ai is a great romance (much better than The Cutting Edge, no doubt), but it seems to be little else other than a romance (albeit a great one, yes, yes). It has the kind of theme that does not get ovations, no matter how good the show may be... Excuse me for not having watched Manie Manie. I could not understand the point of To-Y at all: it seemed to me to be the story of a bunch of guys who tried to be cool, and, like those in The Cutting Edge, I personally find it hard to take these kinds of characters seriously.
To me, Mimi is not just an anime I enjoyed, it is an anime that I enjoyed and will remember.