Professional: When one is an expertise in all areas of a certain subject
Advanced: Being high in level of a specific subject, not a novice
If we look at doctors for instance we notice that some do indeed specialize. You do find that in the professional piano world too, some people specialize as teachers, concerting pianists, music administrators, composer, conductors etc etc (and we have even deeper shades such as someone who is a performer but specializes in a particular composer(s), or a musical therapist working with Autistic's for instance ). Some do more than one of them, some like to focus their attention into one area, there are no rules.
I know many musicians who earn good money from teaching and doing gigs but they have a different day job. Some even play as well as professionals who devote their life to their work, so it is difficult to consider where professional/advanced meets. The consideration doesn't really cross my mind when I am listening to someone play the piano (whether they are pro or not). There was no difference in what music felt to me when I took it professionally compared to when I was doing it next to my non-music schooling and university. The relationship you have with music has little to do with your profession, that is what I ideally like to believe anyway! When I was finally able to devote my time to music nothing really changed, things got harder I guess rather than easier, the path opened up and I specialized in things that interested me (performance, teaching, writing) and broadened my horizons. But the love I have for music didn't change, it is essentially the same, and I think that is the important aspect of music which should drive your work, not whether you are a professional or not.
For most of us professional musicians, music is such a force and influence in our life that it distracts us while we do other work. I noticed this a great deal while studying 3 years of engineering and computer science, although you get through that work and align yourself to work in something non-musical the music is just always at the back of your head. When I would walk around university in my head would be playing piano music and I would be lost in thought (often I would catch myself walking to the music in my head), then when I arrived to my next lesson I had to tear myself away from the music and work again. I would sit in the library listening to piano recordings through my mini disc player (new and wonderful technology back then!), at the end of the day when my uni work had been done I always rushed over to the piano practice rooms and spent a couple of hours or so playing.
Many times people asked me where I was studying music when they observed me playing piano while I was an Engineering student (I played at a lot at lunch time free student concerts Ken Hall Theater on a nice Boston concert grand). When I told them I was not doing a music degree they where always quite surprised, I really didn't take notice of it. When I attended a master class from a world famous concert pianist he asked after I played for him what I was doing with my life, when he heard I said Engineering I noticed him slump in disappointment, I certainly took notice of that! That was the point in my life where I said, "Ok fine this is enough already! Let's take a risk and just do music instead!" and that year I did my first solo concert which was the most successful and still is the only piano solo concert ever to hold a capacity audience in my city's concert hall (which gets international performers from all corners of the world). So in my instance I went from a "advanced" pianist to "professional" in a split second pretty much.
I really didn't feel any different at all, there was a huge amount of more time devoted to music which made things a lot easier, also it made things a lot harder. As musicians we are our own business, we need to constantly push ourselves or we can stagnant and remain in the same position for the rest of our lives. This is ok for some, but for me at least, I always like to grow and change. Taking music professionally I think we consider our future musical path in a lot more detail, it is very easy to get caught up over certain avenues of music, for instance I could keep busy teaching for the rest of my life and do it quite happily, however, where do you go from teaching? A professional can merely teach students in their "little box" for the rest of their life. A business minded professional teacher however will expand their horizons and learn how to expand. It all happens in steps, you move towards what you want you don't just get your end product immediately.
For my teaching experience, it started before I took music professionally as I taught people who came to my parents home. I learned about how other people perceived events and learned as a whole, also could use techniques taught me me but refined in a way which I thought was easier to understand for the individual student. Already my style of teaching was developing, that was to be flexible and yield to the students needs. I had many experiences with teachers who neglected this and they for me where the worst teachers, those that treated me personally I adored and still have never forgotten about.
When I got my drivers licence I then earned extra cash teaching peoples at their home. I learned how to market my teaching services, also it opened me up to many different types of people further broadening my understanding on how people learned. I had many difficult experiences like meeting students who could sight read better then myself, students that didn't pay or cancelled lots of lessons, teaching children who where forced by parents to learn the piano etc. I learned about how unexpected business can be and how you need to plan for the worst. I was also revealed how much further I had to develop my teaching craft to satisfy a more advanced/difficult client. Being able to go out and teach random students from the public was really frightening because there where no safety guards that I had when teaching people that knew me. I took great care with my clients ensuring that I worked out exact what they needed and often I would spend a lot of time at home puzzling out how the next lesson would have to be set out. Structuring lessons for all sorts of different people and needs was really a challenge but as a professional I would work outside of lesson time carefully preparing my lessons and trying to solve my own questions about how a particular student worked.
When I became a professional musician soon after that the next step I took was to teach at a music school where people came to me (there I learned about how to teach with restricted time and teach long connected hours without any breaks.) I learned how to work with other music teachers and school administration/management, something I didn't really have to worry about when I taught on my own where I was my own boss. There where rules and regulations as to how I had to teach/treat my students which I had to conform to. I had to remain flexible to my approach, I would often call my students on the phone and email them, but with students from school I was not allowed to do so which to me is ridiculous since it merely makes the piano lesson a short weekly affair and not something where you converse with your teacher outside of lesson time.
From understanding how a music school is run by being a part of its function I then planned my own music studio and school idea. It is too vast to really point out how I managed to get several grand pianos, a location, build a studio etc etc (having family and friends with a passion for music and disposable income to support the cause certainly helps a great deal, we are blessed in that respect more so than others). Importantly you need to be able to make your own money and invest in your own projects, but certainly it helps to get investors who are excited about the same thing as yourself. As professionals we tend to be open to musical opportunities a lot more and seek people interested in our cause without hesitation. I can relay one missed opportunity in my life before I was a professional pianist, a Yahama piano dealer was in a major shopping center in my city and I asked if I could try out some of their pianos on display. After playing he gave me his card and asked if I ever performed in public before in concerts, when I told him I did he said next time I do one please let Yamaha know how they can help. These connections are really golden and as rare as gold. I was too young at that time to fully realize it and lost the card......
We could even consider advanced pianist to professional like a human relationship, a de-facto relationship moving into marriage. My aunt and uncle lived together for over 20 years before they got married, and the actual marriage itself didn't change anything for them! I feel the same remains for taking music as a profession, nothing changed with your relationship, but now that you have more time for it you can start developing it further (but what is important is that you actually have something to work with at the start, you don't create it from scratch becoming a professional!).