The ending of each term implies something different. Just based off the words, caregiver implies one person in a position of strength and control giving to a weaker, more passive recipient. The caregiver is in charge and takes action. Care partner implies shared responsibility with each person contributing their strengths to meet a shared goal. Most people hate the idea of being a burden and lacking control. Given the choice, most would prefer to think of themselves as a partner rather than a passive care recipient. Care partner is a more recently used term that brings important nuance to complex relationships.
Having said that, it’s important to know and use both terms.