Notifications are not specific to any desktop email program, but rather simple text-based emails sent from a generic SMTP server. Notifications via email are completely separate functions from the Outlook Connector features.
Outlook is used as a separate application and the Outlook Connector allows tasks from the Project Insight portal to also be listed in Outlook tasks. It has the same basic functions as the My Insight page in that it can list tasks I need to work on now or in the near future. Outlook tasks can be less functional than the My Insight page because they are limited to functions within Outlook which were not specifically designed for project management. For example, Outlook doesn't know the difference from a task assigned to you as a resource or a task owner because there is no such distinction in Outlook tasks.
No tasks in Project Insight can show up in Outlook without doing the synchronization to retrieve them as Outlook tasks. The tasks must further conform to the rules of the Connector settings to show up as Outlook tasks, just as in the My Insight page. Using the default settings as an example, only incomplete tasks assigned to me as a resource and scheduled to start in the next seven days will show up as tasks for me in Outlook. Most resources have no reason to look at anything else.