There are several different contexts in which balanced might apply to a DAP.
1) At the level of headphone compatibility - The drivers themselves used in headphones are basically balanced; it's just inherent in how they're designed. Obviously you can connect headphones wired as balanced to the output of a balanced amplifier with no problem, and you can connect unbalanced headphones to an unbalanced output. You can also connect a pair of "balanced" headphones to an unbalanced amplifier output with a simple adapter cable. You simply connect one wire on each of the drivers together and wire them as a common ground connection. However, if a pair of headphones are wired as unbalanced, then they have three wires, which includes a common ground for the two channels. You CANNOT connect this to a balanced amplifier output. All four output lines on a balanced output carry signal, and none are grounded; if you attach a balanced output to a common ground load you are shorting two of the amlifier outputs together.
2) At the level of the amplifier - Assuming the amplifier is really balanced, your stereo amplifier actually is a fully differential amplifier, which means that each channel is really made up of two amplifiers cnnected in a bridge configuration. This gives you the benefits credited to any fully balanced amplifier - mainly cancellation of certain sorts of distortion. There are certain reasons why avoiding a common ground connection on the headphone cound conceivable have some really minor benefits, however the main reason most people get balanced headphones is so they can connect them to a balanced amplifier. Because the distortion on most reasonably well designed modern amplifiers is already extremely low, it's doubtful that the reduction in distortion would be significant.
3) At the level of the DAC - Since any digital audio player with an analog output includes a DAC, considerations about DACs are automatically included in this discussion. Most modern DAC chips have a balanced stereo output. This means they have four outputs: L+, L-, R+, and R-, each of which is actually output by a separate DAC. As with a balanced amplifier, the two DAC outputs for each channel are out of phase, and contain certain distortions that cancel out between the outputs in each pair. if your device has a balanced output, you're going to use all four of those output signals. However, even if the output of your DAP is unbalanced, if you want the lowest distortion, you're going to use all of them. If you connect the R+ line to one input on your right channel amplifier you will get a certain amount fo distortion. However, if you connect both the R+ and R- signals to the two inputs on an amplifier with a differential input, some of the distortion will cancel, and the overall distortion will be lower. There are two things worth knowing here. First, the distortion numbers we're talking about are very low... Second, ALL op-amps, and almost all other modern amplifier designs, have differential inputs, so doing it this way does NOT make the design more expensive or more complicated. It is literally a matter of connecting a few more wires, rather than not connecting them, and about 20 cents worth of additional parts. Now, in addition to all this, most DAC chips allow you to connect each stereo DAC chip to provide a singel monaural output. By cross connecting the all four outputs to a single amplifier in the proper way, you get a slightly lower level of distortion, and typically a S/N that is between 3 dB and 6 dB better. Again note that we're talking about specifications that are very good to begin with. The drawback is that, since each DAC chip only serves one channel, you need to use two separate DAC chips for the two channels. Because the DAC chip is one of the more expensive parts in your DAP, and can cost as much as $10 each, using two rather than one tends to raise the price. (However, regardless of whether you expect it to be audible or not, the best performance specs with a given DAC chip will usually be had with "two separate DAC chips connected in fully differential mono mode". Also note that many DAC chips, designed for surround sound equipment, include eight channels rather than two. WIth these chips, you can have two complete sets of multiple DAC channels, each set connected in differential mono mode, or even cross connected in sets of four, and all on the same single DAC chip.)
The real bottom line of all this is that the performance of any modern DAC chip is extremely good - so we're talking about really tiny differences here.
Likewise, there will be a measurable difference between the distortion spectra of balanced and unbalanced amplifiers, but both are really low, so it's doubtful that those differences would be audible.
(Odds are that, in most cases, other differences in the design of a DAP will make a much more significant difference in the sound than the DAC chip they use.)
I personally don't bother with balanced headphones or headphone amps.
HOWEVER, if you're purchasing headphones, and you have any intention of purchasing a balanced amplifier, then the compatibility issue is worth noting.
If your headphones are wired as balanced, you will be able to connect them to both balanced and unbalanced amplifier outputs, with - at most - an adapter cable.
But, if your headphones are wired as UNBALANCED, then you will NOT be able to connect them to the output of a balanced amplifier.
(There are a few amplifiers that support both - but, when you connect unbalanced headphones to them, they are NOT functioning as a balanced amplifier.)
balanced output was initially brought up in the thread asking about DAPs not DACs. so IMO the situation is different from your example. practicality and compatibility are really the last reasons to go balanced on a DAP.