I like cats rather than dogs
Which, if any, of the above are correct and what is the difference between them?
These are all grammatical.
Sentences 1 and 2 sound almost the same to me, but the latter may be slightly more emphatic. They mean "I like cats more than dogs." When the speaker likes both dogs and cats, but has to tell which he likes more, these are the natural choices.
Sentence 4 means "I like cats, but not dogs."
Sentence 3 is semantically closer to Sentence 4, but sounds much milder and euphemistic. It sounds as if he were saying "I don't particularly like dogs, but I like cats."
So I think Sentence 3 is the one that is closest to "I like cats rather than dogs."