My bet is that a spear thrust from horseback would shatter the spear rather than penetrate plate, although the armor wearer is still gonna be quite unhappy because of the impact (don't quote me on that, I don't claim to have any level of expertise on the matter).
Piercing through chainmail, even higher weave density mail, sounds at least more credible, but debatable nontheless. It isn't like armor was designed to resist slashes and do nothing with other, not edge-based cutting weapons - you still have to overpower the strength of linkage, then the gambeson underneath, yada yada.
I misspoke somewhat about spears through plate armor on foot, the likelihood of you penetrating and killing a knight as a spear wielding footman is incredibly low (praying for cheap armor), but you will be able to leverage the main disadvantage of plate armor, the fact that it deforms and can be made to be more cumbersome than it is protective. A spear driven as deep as possible into a plate cuirass may only penetrate 1mm through the armor, and of course because of the gambeson not reach the knight, but now the knight has a spear driven into his armor, giving leverage and maintaining distance. A similar concept to the pilum, I've shared a video of what they do to shields here previously. (
Edit: Didn't refresh page, vid has already been posted) You won't be fighting much with spears lodged in your armor, and taking time to remove one could get you killed.
In regards to spears or lances off horse, they absolutely would penetrate plate armor and kill the wearer. There is a reason for jousting tournaments the lances were blunted and the armor extra heavy, the force generated from a horse at full charge will give a small spear point deadly penetration, at the very least crushing the armor and leaving the knight unable to move, if not outright killing them from the trauma and broken bones or a quick death from being hit in the head.
Yes, penetrating chainmail is not easy, but very much possible. Even an untrained spearman can deliver a fatal stab through chainmail if their spear has an appropriate thin head and not a broad leaf blade. Once you've made it through the mail, the gambeson is hardly a factor, as the rings hold the spear straight and basically give you leverage to push deeper.
In game, I really wish Taleworlds would be more ambitious. Give us different TYPES of armor that are better or worse vs certain damage types, or more damage types that react differently against higher/lower values of armor, preferably both. Hack damage for axes, as an example, could have higher armor penetration stat up to a certain armor threshold than slash damage for swords, but slash damage gains bonus damage when they deal damage exceeding armor threshold by a certain number. Blunt and pierce could be alternately better vs different types of armor, blunt doing more consistent damage across armor types, but dealing lower value damage, while pierce could possibly completely negate armors such as cloth, leather and cheaper mail and deal higher damage than anything except high value/two handed axes until you reach lamellar and basic plate (coat of plates for Vlandia) armors, where blunt would deal the most damage not accounting for horse speed bonus, which should be higher for pierce weapons.