Why do caregiver and caretaker mean the same thing?

They don't mean the same thing and they are used completely differently. The small overlap in meaning is "provider of care," but their domains are almost entirely separate. caretaker, a 19th-century word, has settled into a semantic space to designate someone who take care of a place. The word that is distributionally most similar is steward. When followed by an "of" prepositional phrase nearly all objects of the prep are places: mosque, cemetery, earth, shrine, etc. caregiver, a 20th-century word, denotes a person who cares for another in need of guardianship or assistance.  When followed by an "of" prepositional phrase nearly all objects of the prep are people: veteran, patient, child, senior, etc. In terms of compositional meaning, caregiver is a straightforward compound, the sum of its parts. Caretaker is explained by the common predicate phrase take care, which precedes the noun--a caretaker is one who "takes care."