polearm master versus shillelagh

I am finding general versus specific rather vague, for this specific case the relevant portion is that polearm master allows a bonus action to deal d4 damage with the polearm or quarterstaff, while shillelagh replaces the damage dealt with a quarterstaff [or club] with d8-2d6 based on level, so which is the "specific" case here?
also in general is there a priority list for when it's not obvious [explicitly stated cases or ones where the alternative seems incoherent]?

edit: apparently the original specific isn't valid, but would the spell still change the to hit ability score or damage type? Forcing that bonus action specifically to still be str rather than the spellcasting ability seems like it would mess up a lot of builds

It's not a matter of specific vs. general in this case; Shillelagh and Polearm Master are doing two different things.

Shillelagh changes the weapon's normal damage die. Polearm Master gives you a special bonus action attack that deals 1d4 damage. That special attack is totally independent of the weapon's normal damage die — it's always 1d4 regardless of what type of weapon it's used with — so it's not affected by things like Shillelagh that modify the weapon's normal damage die.

Per the Sage Advice Compendium:

The benefit from Polearm Master applies to the opposite end of the weapon and always uses a d4 for damage rather than the weapon’s normal damage die. This is true for a Quarterstaff enhanced with Shillelagh just as it is for a normal one.

In the real world, a shillelagh is a stick with a big knob on one end. So the spirit of the rule is probably that the spells adds a big magical knob to one end, not both ends. That said, letting the Pole Strike using the Shillelagh die would only be between 2 and 4.5 average damage per bonus action; that's not the biggest deal, especially since the combo is pretty limited by quarterstaff being a less-than-ideal polearm and Polearm Master not being very compatible with caster classes.

Quote from kenclary >>

Per the Sage Advice Compendium:

The benefit from Polearm Master applies to the opposite end of the weapon and always uses a d4 for damage rather than the weapon’s normal damage die. This is true for a Quarterstaff enhanced with Shillelagh just as it is for a normal one.

In the real world, a shillelagh is a stick with a big knob on one end. So the spirit of the rule is probably that the spells adds a big magical knob to one end, not both ends. That said, letting the Pole Strike using the Shillelagh die would only be between 2 and 4.5 average damage per bonus action; that's not the biggest deal, especially since the combo is pretty limited by quarterstaff being a less-than-ideal polearm and Polearm Master not being very compatible with caster classes.

This is one of the cases when RAW/Sage Advice is silly. The staff is probably thinking all Quarterstaffs are wizard staffs and have only one striking end. This is, of course, not true and there are fighting styles around using both ends of the staff for striking effectively. Powerful strikes are made by shifting to a two-handed, almost sword-like grip, but that is already modeled by Versatile and one of your hands would be at the far end, not leaving much of an "other end" without shifting your grip drastically.

However, DnD is not a simulation engine so don't expect the rules to model reality.

That said, kenclary quoted RAW and the impacts of ignoring it. It's up to the table to decide.

Quote from JamesP_DND >>

but would the spell still change the to hit ability score or damage type? Forcing that bonus action specifically to still be str rather than the spellcasting ability seems like it would mess up a lot of builds

Yes to both.

Shillelagh:

you can use your spellcasting ability instead of Strength for the attack and damage rolls of melee attacks using that weapon

. . .

If the attack deals damage, it can be Force damage or the weapon’s normal damage type (your choice)

Later on, when you use a Bonus Action to make an attack in the manner described by the Pole Strike clause of the Polearm Master Feat, you do this:

make a melee attack with the opposite end of the weapon. The weapon deals Bludgeoning damage, and the weapon's damage die for this attack is a d4.

The "normal" damage die of the weapon is currently a d8 as per Shillelagh, but the weapon's damage die for this attack is a d4.  This attack IS a melee attack using that weapon, so you can use your spellcasting modifier instead of Strength as per Shillelagh.  The weapon's "normal" damage type during this attack is currently Bludgeoning, as per Polearm Master, but if the attack deals damage, it can be Force damage, as per Shillelagh.  So, in this situation, on a hit you can choose either Force or Bludgeoning damage.  Even if somehow after casting Shillelagh but before using Polearm Master you somehow changed the damage type of this weapon to Piercing, for example, you would still be choosing between either Force or Bludgeoning damage in this situation.