Is their much of a difference from the sound coming from a phone instead of a computer mother board with a pair of headphones?
The quality of the AMP/DAC will be lower in the phone.
The files are digital, so in one sense no, but the phone will not decode the sounds as accurately so yes.
The reality is a PC will likely be in a quieter place with less outside noise so over all listening quality will be higher compared to walking around with your phone outdoors some where. So even though there will be a difference the setting will do far more to change to sound than the device.
You can always buy or build am external AMP/DAC to use with your phone if you want better sound from it but for the reasons mentioned above you will probably end up with lower quality sound anyway by the time it reaches your ears.
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Difference - Yes
Noticeable - depends
Phone may struggle to play music through high end headphones (due to resistance) and as @Zibob said - the quality of the DAC/Amp will make a difference. Having said that, basic earbud headphones I doubt you would notice.
My PC has an ASUS Xonar Essence STX and I use my AKG K712 Pro's and can tell the difference between that setup and my phone (Lumia 950) with midrange Audiotechnica Earbuds
ALSO - file quality will matter - not getting into FLAC vs 320kbps mp3 - but crap files sound like crap no matter what you listen with.
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The only difference is the dac and what the dac is capable of.