He prefers dogs to cats.
This merely states a personal preference: He likes dogs more than he likes cats. It does not give any hint as to why dogs are his preference. It does not suggest that he has dogs or cats or is even interested in acquiring either of them.
He would rather have dogs than cats.
The meaning here depends on the meaning of "have", which can be subtle:
The following suggest that there is a specific reason for this statement:
(i) he already has dogs and that he had dismissed having cats.
(ii) he is considering possessing some dogs or cats and that, of the two, dogs are, for him, a better choice,
(iii) he has dogs and cats but sees the cats as less pleasing.
(iv) he has dogs and cats and has been told that he must either get rid of the dogs or the cats, and he has chosen to get rid of the cats.
(v) Cats are making a nuisance of themselves in, for example, his garden and he thinks that if he had to make a choice, he would rather have dogs doing the same thing, or
The general preference is
(vi) In a questionnaire, he was asked "If you had to make a choice, which would you have, a dog or a cat?" He replied that he would have a dog.
This is the same as "He prefers dogs to cats."