Okay, I literally can't watch the embedded video, but to the best of my knowledge, this is the entymology of the different terms:
Foulspawn: A humanoid creature transformed into a grotesque Aberration by exposure to the energies of the Far Realm. Originated in 3rd edition, to the best of my knowledge.
Elder Evil: A god-like Aberration or unique Outsider, possessing apocalyptic-tier powers and influences, specifically created to invoke the "feeling" of Mythosian-style deities in D&D. Originated in 3rd edition.
Starspawn: Sapient stars & planets with ambiguous ties to the Far Realm - being specific, Starspawn refers to the unique entities made by these sapient stars and unleashed on the mortal world, but it is also used for the stars themselves, such as Hadar and Allabar and Atropus. Originated in 4th edition.
Great Old One: Nebulous term that encompasses both Elder Evil, Starspawn, and actual migrants from the Cthulhu Mythos. Originated in 5th edition.