Several things are going on. The cabs own frequency response colors the sound. Another is known as the Fletcher-Munson curve. Your own ears have a frequency response curve that changes as things get louder. That's why when making a patch at bedroom levels sounds different when play it at gig levels. And its not an absolute change and differs slightly with different people. Also the room, the type of cab, your proximity to the cab, the angle your are to the cab the height of the cab, your height, , the mic you're using for the headphones, the brand of headphones, etc., etc. There's so many things I often wonder how I even get a patch to sound good. Let alone in more than one situation. The main thing is to know it will be there, not let it frustrate you (too much) and just do the best you can. Going from a power amp/cab to a pair of headphones next to your ears is bound to cause some differences. Getting something to sound the same from your power amp/cap to your headphones/PA will be difficult at best and just requires some old fashioned tweaking. I use the Fletcher Munson curve and the frequency response of my cab to help but it's not definitive. It's even sometimes better to start from scratch again.
One HUGE guess in your particular situation is that many guitar speakers frequency response drops off significanlty toward the higher frequencies. For example the much used Celestion GT-75 has a major drop off at about 8K or so. Since you created the patch with the cab, when you use a relatively Full/Range Response thing like the headphones or PA, those high frequencies that had been attenuated by the speaker, will be boosted quite a bit.This sounds like part of your problem. I hope this makes sense. I'm not in a position to get too in depth about it.