Both function in a very similar way, but there is a slight difference. Templates are static public boards that members of the Trello community have made for others to use. These are easy to copy to your workspace, and the creator pre-fills some columns and cards with useful information on how to use them, similar to a Word or Excel template.
On the other hand, a public board is an active Trello space where invited members fill in cards and move them around. It’s essentially peering behind the organizational curtain of a business, developer or publication.