From Circle of Spores:
At 14th level, the fungal spores in your body alter you: you can’t be blinded, deafened, frightened, or poisoned, and if an attack is a critical hit against you, it doesn’t deal its extra damage to you.
A poisoned creature has disadvantage on attack rolls and ability checks.
The druid cannot be poisoned: check.
From Damage Types:
Different attacks, damaging spells, and other harmful effects deal different types of damage. Damage types have no rules of their own, but other rules, such as damage resistance, rely on the types.
Does the ability give damage resistance or immunity to poison damage: no.
Contrast this with Dwarven Resilience:
You have advantage on saving throws against poison, and you have resistance against poison damage.
This is nice and clear compared to the UA stuff - but UA is often poorly drafted: maybe deliberately, maybe not.
The poisoned condition and poison damage are not the same thing. It may make thematic sense that someone who cannot be poisoned should not take damage from poison but if it doesn't say it it doesn't happen.