Desktop units typically have:
1.) Higher gain
2.) Cleaner power stage ( lower noise )
3.) More room to separate components ( ventilation, noise, layout, ease of fixing )
As for this statement:
"As far as iv gathered, an amp connected to an external power supply, i.e, the mains, will provide the proper voltage swing needed for dynamic range in the "packed" bits of music. I mean when theres like a crescendo or a climax where all the instruments are playing together."
It really does not make sense is only true under certain circumstances. Sometimes portable amps can sound as good if not better than desktop units. It depends upon implementation of components, noise, power layout, solder joint quality, gain, and output impedance just to name a few.
An issue with battery powered units is as the battery voltage lowers the various bias voltages become marginal on some components and the amount of raw voltage available also lowers. This can often lead to higher noise, lower gain, and other detrimental effects.
If you have a clean power supply that delivers the correct input voltage to your CMOY amp then, yes, you may hear / receive some improvement. Unless you have a good understanding of electricity you may want to do some research or at least talk to someone familiar with building / installing power supplies before blinding trying to plug in your amp to an external power supply unless of course the amp was designed to do so in the first place.
In the $150 range I like the Total Bithead from Headroom. It is built really well, sounds good for the price, and comes with a built in DAC which will improve almost any computer listening experience. It can be powered via USB or battery.
Check it out here:
If you can I highly recommend saving up for a separate DAC and Amp much like the Micro Stack offered at Headroom.
Micro Stack Package
Yes it is relatively expensive, but the reviewers over there at Headroom us the combo to do their headphone reviews making a tried and true combo. I personally like the sound from the combo as well as the price for performance value. They look nice together and take up very little space.