First, "snowball" may be one word, so the dog's name could be Snowball.
Then, while the first statement is grammatically correct and it makes sense, the second has some issues. The way you wrote it suggests that the dog itself gave a new name to snow: "ball". But generally speaking dogs do not name things, or at least they do not do it in English ;-) Otherwise, one may see this as a fragment from a sentence that never got finished. For example: "My dog named Snowball wanted to go out". So probably you intended to say something else but the grammar betrayed you in the second sentence.
Short answer: the two sentences are not equivalent, and likely the second one is not okay.