IMHO nothing, except for some laziness due to the shorter processing time (hours instead of weeks) – and something about fatty acids (see below). I've made M&P base with good success from cold-process soap (though I'm with KimW in that I have no comparison with industrial M&P base):
I finely grate the soap, mixed it with the solvents (polyols, sugar, water), let it stand for a few days (solvents diffuse into the soap), and then gently warm up in a water bath until dissolved/molten. The point of the “soaking” is that you don't need to fully heat it up nearly to a boil (as you would with HP), but the plasticising properties of the polyols will jump in at much lower temperatures, and help fluidise the batter. Replacing heat by time.
Mixing the solvents into the lye at the very beginning of CP might sound tempting (keep in mind DeeAnna's “molecular level mixing” argument!). But the polyols dilute the lye (= slower saponification rate), and (more severely) they liquefy/dissolve trace (we want this, but not that early!), so that the batter will notoriously separate into lye+polyols and an oil layer. Source: own experiment (I nearly lost patience after three days of repeated blending and re-heating a pure palm M&P, and the result was impressively mediocre in the end).
Well, one thing that certainly prefers HP for M&P soap is the fatty acids. You'll notice that M&P recipes are generally very high in saturated fats, and many have pure fatty acid in them (stearic, palmitic). These are needed to get the bar hard at room temperature, instead of sticky/gel-like. But it also means that you'll have to heat the oils considerably to melt everything up (stearic acid Mp 69.3°C). And from there, it's only a small step to a full HP protocol.