One title for the navigation and one for the page (or post) itself.
This is an extremely simple plugin weighing in at 31 lines long.
If you add a custom field to a post or page called "custom_title" then the value of that custom field will be used as the post or page's title when the post or page is being displayed. For example, you have a page called FAQ which will appear in the nav menu as FAQ and you add a custom field custom_title with "Frequently Asked Questions" then the page title will appear as "Frequently Asked Questions" on the /faq/ page, but in the menu it's still FAQ.
This plugin was inspired by the WP Title-2 plugin but uses a much simpler approach. There is no menu or interface of any kind. You simply add a custom field to the posts you want to modify.