People have been asking this question for centuries now, and the answer is usually Bach.
Bach is beloved of performers, because his music can be performed over and over, interpreted a thousand different ways, and never lose its beauty. When you know a piece of Bach well, you begin to collect recordings of that piece, because every recording brings out a new idea, a new experience that you've never heard before.
Beethoven is more beloved of non-musicians. His works are more varied, and more accessible to the casual listener. To a casual ear, all Bach sounds the same, and all Beethoven sounds different. Beethoven's works are much more controlled; that is, there are usually only one or two ways to perform each piece correctly, and one can spend a lifetime trying to perfect a single interpretation of his work. Once you know a single recording of a Beethoven piece well, you may hear another recording, but you cannot help but compare the two recordings and judge which is better. With Bach, any number of interpretations may be valid. With Beethoven, one is always better than another.
Also, there is a quality to Beethoven that I hesitate to describe. It comes across as a sort of neediness, an egocentrism. He always seems to want it just a little too fast, a little too loud, a little too pretty, a little too delicate. It's very self-aware music, as if it was designed to win competitions. There is none of that in Bach, ever. Bach wrote for God alone, and one gets the sense that he never doubted that God liked his work.
If you listen to someone play Bach, you hear someone playing Bach. But if you listen to Beethoven, you hear only Beethoven, not the person playing.