customer care vs. customer support?

I noticed that most companies use the term ‘customer support’ while some use ‘customer care’. Is there any difference between both? To me, ‘customer care’ somehow conveys the idea of ‘care’, ‘being constantly there for the customer’ while ‘customer support’ means that you can get help only when you really need it.[YSaerTTEW443543]

TOEIC listening, talks: A new company executive introduces himself to the staff[YSaerTTEW443543]

I think “care” is a more touchy-feely and personal word that is supposed to make the company sound more, well, “caring”… In reality, I don’t think there is much difference in meaning.