It's complicated and if you have to deal with this professionally, you should seek advice from someone (accountant/lawyer) who knows the system.
There is such a thing as a 0% VAT rate (but not in every EU country). If you are selling goods taxed at 0%, you are entitled to reclaim the VAT you paid on your supplies and have to keep track of it accordingly. More information on this from HM Revenue & Customs in the UK.
There is also such a thing as being VAT-exempt. In that case, there is usually some other specific legal mention to include on all invoices. If your whole business is outside the VAT system (e.g. very small businesses and non-profit organisations), you do not itemize it on your own invoices, pay VAT on your supplies and cannot reclaim it. If you sell goods to such a business, you charge them VAT just like you would in a transaction with a private person.
So both exist and are distinct.