Same difference.
Not really. Club statements just "say" something; press conferences are a song and dance.
For example, you wouldn't call a press conference for this, but you would issue a statement - http://www.evertonfc.com/news/2017/04/15/statement-from-everton-football-club
So the statement could easily be "After protracted talks with Ross Barkley, we can confirm that the player will not be signing a contract extension with Everton at this time."