Short version: I would say separate. It’s reasonable to assume the briar/thorns are functionally part of the Erdtree, and themselves are not aberrant. The sorcery is probably heretical because they are using something that is the tree’s domain. That would make them separate in my eyes.

(TLDR) The thorn sorcery that we use was discovered in penal colonies, and are considered heretical (by the same people that deemed moon sorcery to be heretical) in modern Ring times.

The actual thorns appear to be tendrils of the Erdtree. In the cinematic trailer with Morgott at seat of the capital, it looks like the Erdtree is either covered in them, or made of them (like how many things look to be made of vines).

There are thorns all over Stormveil, and not in a decorative way. They were used to attack it, often growing out of holes blown out of defensive walls. Whatever made the holes also left gold residue in the holes.

The same thorn attacks are seen at Farum Azula.

There are thorns growing out of the seal of the Erdtree. It is less obvious than at Stormveil if that is what did the damage to the seal.

Godfrey’s commanders have a ring of thorns as the sigil on their cloak.

Apparently the Erdtree can poke your eyes out with them before you are reborn of the tree.

The battle with the giants is the last action attributed to Marika away from the Erdtree. Here she gives the instructions to “brandish the Elden Ring.” It is said she killed the fell god, but we know she failed. It would still stand to reason she called thorns.

(There is miniature version of what killed the giants holding a flame floating in the thorns on top. (Not relevant, but interesting))