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Both Buddie and Woodsen are in the 2020s, while Woody's world is in the early 2000s. The only difference between the two is that Buddi enjoys vines and memes (Something he got from his Andy and Molly) while Woodsen is more concerned with his body count.
(P.S. Woodsen got the knowledge from his Forky, who will sort of act like a stand in for Jake Wheeler.)