A public board is a board with a unique URL that is very hard to guess, it's secure if you don't share with people that you don't want to have access to it. It's nice for events with a large number of people and that you don't want everybody to create an account. To share with people you just need to send them the URL of the board.
Team boards are better for a fixed group of people that does recurring meetings together. The people on the group usually don't change very often and the boards are more secure. A team board is a more secure board, only people that you invite will have access to it, you will be able to add or remove people.