"Customer care" seems to mean looking after customers, whether this being making sales or after-sales service.
There' another meaning of "customer experience" that is very similar.
It is a relatively new term for the overall experience that customers have engagging with a company or organisation.
So "customer experience" is the style of how they are treated.
It tends to be used for the design and management of the experience that customers get, over all. As a result, "customer experience" tends to be a term for a biger thing, a cultural design or decision of an organisation. Hence tends ot be more for large organisations. Often used as a justification for everyone in the organisation having to do thingsin the same way, so the customer gets a uniform, same everywhere, customer experience.
Some might think that "customer experience" is a grandiose term for just doing the right thing, what should just be good practice, but making it sound like an art-form, but few would be brave enough to say this in an organisation that has a department called "Customer exxperience" that management believe in.