Allow user to Edit Posts but not Add New?

You will have to do something like this:

function hide_buttons() {
    global $current_screen;

    if($current_screen->id == 'edit-post' && !current_user_can('publish_posts')) {
        echo '<style>.add-new-h2{display: none;}</style>';  
    }
}
add_action('admin_head','hide_buttons');

See: http://erisds.co.uk/wordpress/spotlight-wordpress-admin-menu-remove-add-new-pages-or-posts-link for reference