Though I agree with Joker III above that tone is basically in the gear (a slowly strummed open E should sound roughly the same thru the same rig whether I or JP slowly strums it), a better/different player may be able to coax the character of the rig out in their playing differently than I in more complex playing, which can make the tone seem different to a degree, depending whose hands are playing it.
Going back to the orignal topic: How to take someone else's preset and make it sound the same for you? From the discussion so far, the detailed methodology that's rising to the surface seems to be:
1-Confirm no post processing on the original tone you heard (if there was, then of course you may not be able to match it without knowing what the post processing was). Recognize that some pp is unavoidable - ie what is introduced by upload to youtube.
2-Listen to the reference tone using the same monitoring environment as you will use to match the tone. IE if you listen to Leon's vid on the tv set, then try to replicate thru Axefx to your Yamaha monitors then you've already set yourself up for possible difficulty.
3-Get the exact dry track, preset, and global Axefx parameters from the author of the tone you heard in 2 above. Also, to mitigate the effect of youtube pp, get the author to send you a direct recording of the tone also.
4-Match the tone using the dry track and tone recording provided in step 3 thru your rig leaving preset parameters the same, confirming no relevant global parameter diffs, and adjusting only for differing monitoring. If you can't nail the tone at this step, then check your rig for differences before moving on.
5-Match the guitar tone. Confirm you are using the exact same guitar including all the parts that can make a difference: brand, style/model, pickups, pots, caps, wiring method, saddles, nut, strings, tuners, woods (resonance), ... The dry track in 3 above may be useful to compare your own dry tone to before moving on to the next step.
6a-Substitute your playing starting with simple chords/notes - ie if it is a heavy distortion patch, just match a few simple power chords. It may be useful to have a few simple chord strums/notes at the beginning of that dry track you got if the tone author can give you that also.
6b-Substitute your playing with original material/riff ensuring that you actually know how to play it well enough to avoid tonal differences caused by playing technique/proficiency issues.
Approaching it very methodically in this way would seem to pretty much ensure success. The guitar matching would be the most difficult step, and one of the most crucial steps given the Axe's sensitivity to different guitars (a tribute to its quality). I'm betting few go thru this type of effort which explains why this "complaint gets raised in so many modeller forums. It's not magic - there are specific reasons for any differences - a
scientific approach, however tedious, solves the mystery.
As noted above, a lot of these youtube dudes are great preset architects so just learning their preset building methods is of great value.