Let me be clear: BOTH of these methods are great and have value. The important thing to understand and recognize at the outset are these two things in THIS order:
What is your learning style? (Visual or Auditory or both?)
What are the contexts you will be playing music in? (personal enjoyment, concerts, accompanist, church, bands, gigs, jazz, improv, song-writing, etc.)
If you don't know the answer to those questions yet, I can help you, but they are essential to know before any kind of lessons can proceed.
In general, classical training focuses on technical skills and learning to read the music notation that others have written (classical composers, hymn-writers, etc.). An understanding of music theory is still very valuable and important, but not required to play classical music. The emphasis here is placed on understanding all of the notation markings and what they mean, so that you can then play the music as it was intended.
By Ear/Music Theory
In general, in this method/style you are focusing on listening and understanding how music is constructed (the theory of music). There is much greater reliance on ear training and being able to recognize musical patterns and harmonizations simply by listening rather than seeing the notation on the page. This method and mode of playing music also lends itself much better to creativity, song-writing, and improvising (making your own music spontaneously).
What About BOTH?
There are those who can live and play in both worlds, but in general, people tend to gravitate toward one or the other. Although I began down the classical road and have a deep appreciation for and have worked with many classical musicians in my life, I am FIRMLY PLANTED in the BY EAR/MUSIC THEORY camp and that is the place I teach from.
IF that is what you're desiring in an instructor/coach, then I invite you to reach out and let's see if I'm a good fit for you!