Publish Object

This allows you to publish the updated content object that is currently in preview mode.  Until you publish the updated content object no visitors to your site will see the changes that have been made. Objects can be pending publication for an indefinite time.  It is ok to leave them pending publication while you seek approval or feedback from others or require more time to continue revising the content.

Publish Page

Publish page is the same thing as publish object, but will publish all objects that require publishing at the same time.  This is useful when you have multiple objects on a page and have updated many of them.