Attacks delivered by spells are magical.
Page 8 of the Monster Manual under "Vulnerabilities, Resistances, and Immunities" states (added in an errata):
a magical attack is an attack delivered by a spell, a magic item, or another magical source
So thorn whip delivers a magical attack dealing magical piercing damage.
Bludgeoning/Piercing/Slashing damage from Area of Effect spells is probably magical and does not matter but with two exceptions.
Consider earth tremor:
You cause a tremor in the ground within range. Each creature other than you in that area must make a Dexterity saving throw. On a failed save, a creature takes 1d6 bludgeoning damage and is knocked prone.
Is this bludgeoning damage magical, or is it mundane - just the damage from you falling down? According to the Sage Advice Compendium, it is magical:
Determining whether a game feature is magical is straightforward. Ask yourself these questions about the feature:
- Is it a magic item?
- Is it a spell? Or does it let you create the effects of a spell that’s mentioned in its description?
- Is it a spell attack?
- Is it fueled by the use of spell slots?
- Does its description say it’s magical?
If your answer to any of those questions is yes, the feature is magical.
Earth tremor is a spell and is fueled by the use of spells slots, so it is magical, though there is room for a DM to rule that the effect of earth tremor is just that you fall over and hit your head taking non-magical damage. The two cases where this distinction will actually matter are explained below.
What resistances do these spells bypass?
For the resistance asked about in the question, damage from spells will ignore it:
bludgeoning, piercing, and slashing from nonmagical weapons that aren't silvered
This resistance is specifically to nonmagical weapons that aren't silvered. Since spells are not nonmagical weapons that aren't silvered, they would ignore this resistance.
On the other hand, the Awakened Tree has resistance to bludgeoning and piercing damage:
Damage Resistances Bludgeoning, Piercing
This does not specify magical or nonmagical, nor does it specify a source, such as attacks or weapons. This is wholesale resistance to these damage types, no matter the source.
Another interesting example is that Demilich, which has:
Damage Resistances Bludgeoning, Piercing, and Slashing from Magic Weapons
So while the Demilich is resistant to attacks from magical weapons, it still does not have resistance to bludgeoning, piercing, and slashing from spells.
Stoneskin/Armor of Invulnerability
Above I stated that magical/nonmagical BPS damage from AoE spells only matters with two exceptions. The first is the spell stoneskin:
Until the spell ends, the target has resistance to nonmagical bludgeoning, piercing, and slashing damage.
The second is the magic item Armor of Invulnerability:
You have resistance to nonmagical damage while you wear this armor.
These are the only two cases where the DM's ruling about area of effect spells will make a difference. In all other cases of resistance to BPS damage, the resistance directly applies only to nonmagical attacks or weapons. In these two cases, it does matter how the DM rules, because these resistances do not care about the damage being from attacks or weapons, only if it is magical or nonmagical. In all other cases, the damage from earth tremor will bypass resistances that mention BPS damage, no matter how the DM rules on its magicalness, unless that resistance is just the unqualified bludgeoning damage, as we saw with the Awakened Tree.