I'm going to cut to the chase, in D&D 5e, if a player wearing adamantine armor gets hit with an adamantine weapon, what happens?
The player is not an object; the hit is a normal hit
Adamantine Armor is a magic item from the DMG. The relevant effect:
While you're wearing it, any critical hit against you becomes a normal hit.
Adamantine weapons on the other hand are described in Xanathar's Guide to Everything (emphasis added):
Whenever an adamantine weapon or piece of ammunition hits an object, the hit is a critical hit.
Since creatures are not objects, adamantine weapons have no extra effect and it's a non-critical hit (even if the d20 roll was a 20).
For using adamantine weapons against adamantine objects (assuming they inherit the property of the armor) there are two contradicting specific exceptions to the general rules, so the DM is expected to use their sound judgement (this also applies to dealing with objects in general). Personally I would lean towards the effects canceling each-other, and would consider applying that as a house-rule to the original scenario, should it ever come up (at least if it's one of my PCs wielding the weapon).