The actual improvement (physicaly and musically) of a piece we memorise never stops. So we could get philisophical and say we never actually completely finish a piece. However I think it is not the right way of thinking because we are actually improving an aspect of our playing as a whole as we maintain contact with our memorised repetoire.
So there are 2 things we constantly improve with our music. That is the PHYSICAL excecution and the MUSICAL expression/interpretation etc. These two things are constantly improving along side one another however they draw upon different main sources for improvement.
The PHYSICAL part improves best with repetition, experimentation, together with the aim of increasing the physical efficiency, improving a more comfortable natural feeling ("our fingers like honey" as Mozart said) or effortlessness of playing the given piece. Our physical action also improves as we absorb and become very familiar with general "procedure" at the keyboard (arpeggio, scales, chord progression, building blocks of music) and understand how to notice, apply or control changes of these standard procedure that we find in a new piece.
I noticed that when a student who for instance never played an arpeggio etude and then starts to learn one, is confronted with a huge amount of discomfort. The challenge to learn to balance playing arpeggios in the hand in this instance is a big brick wall and hinders their development. However after a while full of struggle and learning, once they somewhat master the basic idea of arpeggio, when faced with it later on down the track they have the facility to deal with it and thus it takes a lot less time in comparison to the first time they attempted it. This would be considered an improvement in the physical aspect of playing in my mind. The faster your rate of absorbtion of new music the better you are becoming. So spending 6 years to learn to play one Liszt Etude is not as "good" as someone learning 60 smaller pieces in the mean time. The exposure to "procedure" at the piano is just not covered in one piece, we have to see many examples from many different composers.
So this means that we must always search for these "procedures" at the piano and make it second nature to us. This is not to say if you cannot play rapid scale thirds in one hand that you should make that your aim, depending upon your musical taste, depending upon the your physical body, and depending on your commitment to your music, different things will be targeted. It is important to understand what procedure you face in pieces you currently learn, this is why a teacher is good to help draw attention to these details. Often we can target and know what procedure is difficult for us, but we have no idea how to deal with it. There is always a method to go about working on your difficulties. Unfortunately one cannot write a paragraph to explain how to practice away your difficulties because each person is different and different things will cause different changes to peoples undertsanding. Again, a good teacher wil help you because they form a relationship with you musically and then know how to guide and offere adviced based upon your needs.
I really feel like a doctor when teaching piano. As I watch the same student play for a few months, eventually we learn to sense when they are playing relaxed and when they start to get uncomfrotable. Each person however tenses differently in different places in a score. Particular fingers are unbalanced, or sense of the center of their hand lost in different places. As a teacher you should notice when this happens and offer advice to try and make them play in their relaxed mode.
I always ask when I see phsyical transfer to the keyboard that looks uncomfrotable or strange, Are you really comfortable? Which finger(s) is the most difficult in this passage, where is the centre of gravity, which finger(s) balance the hand? These questions have deep answers which give reason as to why we do this or that to our hand in a given passage and also very various depending on the persons fingering and their physical hand.
Sometimes we have to SHOW what relaxed playing looks like and what tensed playing looks like. We have to be actors and OVERDO the students flawl. It is amusing because I always do it over the top and the students laugh. But it highlights where they are going wrong effectively.
This idea of being shown how it is done is an important observation but not as important as actually physically practicing it yourself. So watching videos of great masters will help you phsyically but not really that much. This is where our MUSICAL ability benefits most. From observation our MUSICAL undertstanding improves and becomes more refined.
I remember when I very first started to teach piano as a young teenager I tried to improve peoples musical understanding and decision making by telling them what to do. Louder here, softer here, faster here slow down there hoping that as I repeat it over and over again with many examples they will learn. But eventually I found it to be very ineffective to improve the students Musical understanding. They rely on the teacher to make the initial decision and if left alone are afraid to make the decision or make ineffective musical decisions (weak dynamics, flawled tempo control).
I found however students who have a particular interest to listen to music and who constantly ask me to give them new music to listen to have fabulous MUSICAL understanding. They have exposed themselves to listen to high quality playing and this only enhances their musical decision making. These students I ask to listen to piano music with the sheet music and follow it, section it up, even if it is a piece you dont wantt o learn yet, just break it apart and notice procedure at the piano and also musical intepretations that the recordings present. These students when faced with music know what sound has to be produced but are phsyically challenged to get there. However they have the clear understanding of where they want to get to which is a lot better than a students who only listens to what the teacher says and cannot make the decision themselves from what they have heard before.