Not enough info. What cable? What headphones? What amp(s)? Too many variables, enough that you'd be comparing apples and oranges.
You have to study the principles, though. A good low impedance amp capable of delivering required power to the headphones for your target SPL that could be configured for either a differential output or single ended output, and the gain difference compensated for, then there would be no audible difference. But if you're comparing two different amps, different levels and gains, and doing the comparison by unplugging from one and inserting into the other, yes, you'll "hear" a difference, pretty much guaranteed. But your "test" is flawed in several ways, and meaningless when attempting to isolate the effect of a balanced connection vs single ended.