Zaten means ''already'', but also it has lots of other meanings like ''besides'', ''anyhow'' etc, so you can use it for connecting the sentences. Neyse means whatever, literally. Ne means ''what'' ise means ''if ever'', so neyse means whatever.

Neyse We sometimes use this as a word to describe "never mind, it is not important, let it go" or something similar. For instanse - Beni dinlemiyor musun? -> Aren't you listening me? -Neyse, sen meşgulsün. Daha sonra söyleyebilirim. -> Never mind, you are busy. I can say it later.

Yes, there is major differnce between them. Ben zaten bir kaleme sahibim = I already have a pencil Neyse, biz devam edelim = Anyway, let's continue