a monitor is a special kind of speaker. a studio monitor gives you the truest reproduction.
most all speaker sets/headphones add some frequencies to the sound in their own attempt to create what they believe is a better quality sound. however, that's very subjective and different for every brand, every quality, etc. Sometimes bad quality speakers/heads can't handle a full spectrum of frequencies. Many times their additions accentuate the main parts of the sound to cover for their deficiencies. A studio monitor will give you the truest reproduction of the sound without any additions, and (hopefully) reproduce the entire spectrum of frequencies. this is very important when mixing/mastering because you want to get the purest, best quality sound before people put their own EQing on it, and/or their speakers play it a bit. For instance, if you go to Radioshack, one set of speakers might have a really great treble, but shitty bass, and another might have creamy mids, but shitty high/low frequencies. that's just the quality of the sound reproduction. A studio monitor's purpose is to give you the truest reproduction of the sound without altering it as said above.
so, in that way, your headphones, depending on the quality, are probably dipping out the bad frequencies. most headphones can't handle a full spectrum of sound, so they kind of cut out the extremes, especially the bass, which is probably why it sounds different in the heads. and seeing as its a cheap ass amp, that's probably why its better