A poleaxe (pollaxe) is basically just a Halberd with a shortened spike anyway (possibly even an evolution of the Halberd?)
Pretty sure neither is an evolution of the other; they're different weapons for different purposes used in slightly different times and places. There's lots of variation on the pole-arm theme, most of them variations or combinations of the long axe and the spear, or agricultural or building tools with hooks and spikes added on. You've got your bardiche, your bec de corbin, your daneaxe, glaives, mauls, voulge etc. all of which were quite specialised but similiar enough that someone who was competent in the use of one generally wouldn't be that bad in the use of another. Iirc (I don't have any reference books here) a halberd would have generally been used by more lightly armoured men of foot against cavalry and later on as a medium range weapon to reinforce pike formations before the full crystallisation of pike and shot tactics. Poleaxes are more of a weapon for combat on foot against other men on foot wearing armour, I think you'd probably find their use concentrated in the 14th and 15th centuries during the period when plate armour use was at its widest. I think there might also be some distinction about the length (poleaxes are shorter?). Also, looking back at that picture, I'm fairly sure those weapons are slightly innacurate anyway because they should have metal strips (the name of which escapes me) bolted down the sides of the pole that attach the weapon head, rather than just being hafted like a spear.
Then again, definitions of these things are hardly rigorously scientific. Even mass-manufactured pole-arms from the same maker would have all been slightly different.