I've never really heard the latter before; and, I'm not really sure if it has the same meaning as the former (I'm not even sure if it makes grammatical sense). I'd appreciate it if you'd let me know if they have any difference in terms of their meanings.
The construction Would you care for [some] X? is a kind of "frozen form" largely restricted to extremely polite / formal contexts. Usually where the speaker (a restaurant waiter, for example) is addressing someone of higher social status). X can be either a noun (something being offered) or an infinitive-based verb clause (an activity being proposed)...
1: Would you care for [some] coffee?
2: Would you care for a cup of tea?
3: Would you care to sit outside?
4: Would you care to follow me?
Note that the first three are all servile / polite offers (speaker will provide you with coffee, tea, or an outside seat if you want it). But #4 is effectively a request - a stylized / polite way of saying Please follow me.
Also note that unless he was being deliberately facetious, if the addressee didn't want coffee, he probably wouldn't reply...
Thank you but no. I don't care for coffee
...because the negated statement I don't care for X is a stylized / dated way of saying I don't like X (ever, I'm not just refusing the current offer). And the non-negated Thank you, yes. I [do] care for coffee simply isn't something native speakers say.
The primary difference between Would / Do you care for X? is we use would in the context of polite / formal offers and requests, where care for means want / like. We only normally use do in contexts where care for means to look after / tend to the needs of (or sometimes to feel deep affection for, but that "romantic" sense is becoming increasingly dated today).