Amps, tubes, DACs, even pads -- all these things have a considerably larger impact on sound than an SE vs. balanced headphone cable will ever have, so I wouldn't expect too much of a difference, if any at all. The difference you'll hear will almost entirely stem from the fact that you'll be listening to a different amp and DAC than the ones you are used to. But as long as you're listening to the same amp and DAC, and all you change is just the headphone cable, I doubt that things would audibly change a whole lot for you.
That said, with certain tracks* and ultra-revealing gear, I sometimes feel that separation is a tad better with balanced headphone cables. Which would make sense, because pretty much all that SE vs. balanced does in headphone cables is to keep the two channels' grounds separate from one another, so there should be less (or rather – and ideally – no) cross-talk.
But then again, it's entirely possible that what I'm hearing is just psychological.
* especially those where both channels were recorded and mixed completely and perfectly separate, as it's often the case in electronica as well as some super old recordings
[As an aside: Don't get "balanced" headphone cables confused with "balanced" interconnects. They may share some terminology, but are two entirely different concepts. If anything, balanced headphone cables work essentially like SE interconnects: one "hot" wire and one ground per channel. Balanced interconnects have two hot wires and one ground per channel, and each hot wire carries the same signal but with opposite polarity. (Well, at least in the case of differential signaling. It's complicated.) Balanced (differential) interconnects can make an audible difference -- especially over longer distances and/or if you have to route them past something that causes a bit of interference, like badly shielded cables or antennas, and data or power lines -- because you can use the two opposing signals to "filter" out any induced interference from your audio signal for as long as both opposing signals share the same induced interference. This is not the case for balanced headphone cables, they're strictly just two SE connections in one wire, one per channel.]