Classically trained pianists have normally developed a very healthy respect for the written symbols on the page. If Beethoven wrote an F# for 2 beats then that is what we should play. In the classical sphere, there has always been great emphasis on learning to read music symbols (the black dots etc). Classical piano music also (on the whole) requires more discipline in approach than when playing pop; its just my opinion. For example; not only must two clefs be learnt to read classical piano music, but there has to be a certain studiousness. Read any edition of the "Pianist" magazine and you will regularly see how articles on various aspects require you to work hard and to approach matters with the discipline and patience of a top sports star.
Pop music works in a very different way: I'm NOT saying pop musicians don't work hard, but they usually don't require as academic an approach to reading written music. Often, for example, pop musicians will work with their own compositions as opposed to the vast majority of amateur classical pianists who rarely compose their on material. Although there is a market for 'cover' bands who specialise in reproducing exactly what Coldplay, Abba, Beatles (or whoever) did with a song, pop musicians usually strive to reproduce their own new songs, style, sound and (of course) image. When they are honest, they admit that they have influences. Very rarely are we able to find music books which reproduce exactly [in treble/bass clef or even guitar TAB] what The Beatles, Genesis, ELO, Take That etc actually sang and played. I am not knocking pop music: it is simply a different way of operating.
Jazz is not simply playing by ear. Playing by ear is not necessarily jazz. True jazz is essentially improvisation in styles largely derived from the African slave communities in the USA and was in full blossom by the 1920s. Its essence could be ragtime, New Orleans, swing, big band and the more art-form-like be-bop to name just a few incarnations. Also, improvising is not necessarily playing jazz: classical musicians have improvised for centuries. Regarding Jazz: I love the work of Brubeck, Shearing, Oscar Peterson, Art Tatum and others; they are amazing and have enriched music wonderfully; but teaching improvisation is not yet a strong point of mine so if that's what you are after, then please look elsewhere. Having said that, if you are happy to work on jazzy/pop pieces that are fully written out on treble and bass then I may be the teacher for you. Examples of such books are found in the "Making the Grade" series, as well as books of pieces by John Kember, Cornick, Norton etc .
Christian praise and worship music is a repertoire I am delighted to help with as I am Director of Music for Parbold Curch in Lancashire. Usually those who are interested in this already attend a church and appreciate that this aspect of their lessons is not helped by a good repertoire of relevant graded resources: often the solo piano arrangements of praise and worship are not very pianistic at all, but it can be an opportunity to develop "playing by ear". Composers include Kendrick, Townend, Tomlin, Matt Redman, Tim Hughes as well as traditional hymns.
PLAYING BY EAR
I can easily help someone to play by ear or to work from treble clef only but that is not the same as pure jazz improvisation. The ABRSM [examination body] actually has a separate syllabus for jazz exams which I thoroughly recommend. But do not underestimate the work involved with this route: it is a different approach: it is not necessarily easier than reading the dots (crotchets etc).