Question

A black and a white dog and a black and white dog tell me difference​

Answer 1

Both structures are grammatically correct, but the former is what you want, I think.

"The white and the black dog" is correct because here, ellipsis just dropped the "dog" after "white". "The white dog and the black dog" is the full or "original" phrase, and it's definitely talking about two (and only two) distinct dogs.

"The white and the black dogs" is grammatical but messy. Maybe it can be parsed as [[the white and the black]AdjP dogs]NP (where "the white and the black" is one chunk modifying "dogs"). That still doesn't say how many dogs there are. Maybe one of each colour, but maybe more. More confidently, I can say it can be read, "the white dogs and the black dogs" (because of ellipsis like above) where we have a minimum of 4 dogs (2 or more white, 2 or more black), or even "the white (dog) and the black dogs" Where we have at least 3.

Answer 2

Answer:

A black and a white dog are 2 different dogs.

whereas a black and white dog is one with combination of 2 different colours.