One of the things we should and do strive for is flexibility and versatility within our instrument and musicality, and while hours and hours of practice get us to a certain level of ability, the "last phase" is a lot less concrete than we'd like it to be. I'd get really frustrated with myself in college, and even now sometimes, because of my inability to make something sound "like my instructor's sound or interpretation." I'd get even more frustrated going into an audition, playing what I thought was a great round, technically sound, and wouldn't get passed the first round or wouldn't win in the final round. I thought that practicing my technique and the pieces at all the tempos and dynamics was enough for my lessons and the audition, but there was always something missing. 2+2 just was not equaling 4 and I did not know why, and it was making me crazy.
Instructors can't directly teach what the "missing ingredient" is, but they can direct you towards what it is for you. You've probably heard these phrases before: "Sing more" "Play it like a vocalist" "It should flow more" "Play it like a dance" "It should sound scarier" "More elegant", etc. The adjectives could go on, but none of them give us a specific answer, at least one that we can process into a technical exercise in the moment. These are all feelings or subjective ideas, but they all produce an emotional response, both in the musician and in the listener. It's basically going for an effect rather than trying to play the page, by playing the page as an effect (that's the way I think of it now). Yes, we have to play the page, but if that's all we're doing, there's no point and it ends up sounding boring. Usually we take this problem and try to solve it by phrasing more and/or having more dynamic contrast. This does help a lot and usually will work, especially in excerpts, but when a solo piece comes up, specifically Bach, there's a certain way of playing his music, traditionally and otherwise, that makes it sound great and not just good. You can't spell out what it is, because it's so subjective and almost indescribable.
If I had to give this a specific description to what it is, as far as what it is to me and what my instructors and colleagues have told me, it's this: technical ability to accomplish (as close to) anything that is thrown your way, letting your technique become like conversational speech (making that technique second nature so you don't have to think about it and just play), playing the effects on the page (if there's a waltz pattern, there's a certain way a waltz feels, same with a march or otherwise), going for your personal musical choices and instincts (not just playing a rehearsed way of playing but always looking for another way, within the structure of that piece). Finding freedom within your technique, within the pieces you play, and within your musicality and instincts of expression (some people say playing classical music like it's jazz, basically like you are making it up, or making it sound natural and "unrehearsed").
As soon as I let go of worrying about playing everything exactly as I rehearsed, or playing perfectly, I started feeling freer in my performances and it allowed me to focus on and make more phrases, even better than I had rehearsed. The same goes for my technique. When I knew I was at a point where I didn't have to work to make anything happen, I accepted that, "It doesn't have to be a battle every time" or "I don't have to do anything to make this work." That's when my technique really started feeling more natural and second nature, and again I could focus on the music more, rather than worrying about what sounds come out. Your body and brain are extremely intelligent and really good at playing off of your instincts, so if you have worked out the technical parts of the piece, just let go, go for the musical idea and your body will make it happen.
It's scary though because we want to make sure things happen in our performance. That's what we practiced all those hours for right? We probably broke everything down, played things through slowly, tuned every single note, listened to a million recordings, but if we don't let go, it will sound like we are trying, or working hard, basically, it won't sound natural. We have to get to a point where we don't have to worry about our technique. If we can produce the music on the page and the phrases we want, then our technique is fine, and we have to be ok with that. Our technique is fine. Now we simply need to let go, let our bodies do what we know they can do, and just go for the music and the effects it asks for. If we've done the specific work on the piece, we don't have to do that specific work in the performance. Just let it go, be free and play.