I am going to admit first I haven't tried enough amps to be able to say for sure, but one of the benefits of the balanced gear I have tried is the wide soundstage. I've also owned a single-ended dual-mono amp that was similar. My impression now, while not fully understanding audio circuitry, is that it has to do with design of the amp what benefits there will be in those choices made by the designer. Balanced is just another design (circuit) choice from the perspective of the designer and their goals, but happens, from the viewpoint of the end-user, be a much more significant one due to the different plug(s) required on the headphones and the increased cost being that the amp has all its circuitry doubled, at least in the case of most solid-state designs.
What would be more important than whether or not to go balanced would be to consider what goals one has with one's system. Buying gear that is higher-end you'd hope would give you more detail out of your music and better performance when the music is more complex, taking away more layers between you and the music, so to speak. This would also have a lot to do with the kind of music you like. A useful question is: Do you feel something is missing from your headphone experience?* At the very least this may relate to something simple, like the amount of bass, yet other times it may be something like instruments sounding less digital. From there it may be easier to suggest what to upgrade next.
But then, this is Gear-Fi and who doesn't want to take a shot at something potentially better for a while?
*Such as speakers? Sorry, had to make that joke.