Exploded archive is a direct copy of your files in your proejct to the app server. This also ENABLES hotsync (real-time syncing of changed files to your app server).
Some app servers do not support exploded deployment (like Websphere and I think WebLogic).
Packaged deployment is a packaged (WAR, EAR, etc.) deployment of your project into the app server. From there the app server can uncompess and deploy the application or some will likely wait for you to login to the console and deploy the app yourself.