Neither do I vs I don't either [duplicate]

I have read in several sources that in agreement with a negative statement one can use either, for instance:

I am not a child, and my cousin isn't either.

At the same time, there is the word neither that can be used here:

I am not a child, neither is my cousin.

From the sentences aforementioned, which one sounds more natural? Which is more formal/informal? Is there any difference at all?

which one sounds more natural

Both are correct, but the second one sounds more natural to me as a British English speaker.

Which is more formal/informal

Neither sounds particular formal/informal.

Is there any difference at all?

Not really. Both are valid, grammatical sentences. The second just sounds a little more natural to me.