The public address is the location on a particular blockchain where details about your transactions (and therefore how much you own!) for that particular blockchain are stored and updated.
The public address is used ‘publically’ to tell other people/exchanges etc where to send that specific coin. It is similar to a bank account number.
The Private Key is only for sending money OUT and should never be shared. It is similar to a PIN code.
Fortunately, each blockchain has its own format for creating the public address and they also create them in such a way as to reduce the chance of a typo being accepted. This however does not make it impossible!
If you give someone the wrong address then it is possible the coin will be sent to the wrong person or even to nowhere, where it will be lost!
Click here for a further explanation of private keys and wallets.