All routing and map data is still in the car with the new navigation. But the routing is done by Tesla software using downloaded map data from OpenStreetmap and not third party navigation software and map data (from Navigon?).
The software is using 3 sets of map data, 3 sets of speed limit data and 2 sets of traffic data.
The console map display is using map and traffic data supplied by Google, used only for the display of the map on the console display - and not used for navigation. Google updates the map data frequently, typically catching road changes within a day or two.
The cloud server uses its own map, speed limit and traffic data. We don't have any information on how up-to-date this information is. It will be much more up-to-date than the onboard data, and possibly not as up-to-date as Google (since data is their business). The cloud server is used for calculating (and occasionally recalculating) navigation routing, unless this feature is disabled in settings or the cloud server isn't reachable over the internet.
The onboard software uses relatively data that is guaranteed to be out-of-date. The map data is typically updated annually and is usually at least 1 to 2 years old, resulting in many discrepancies in areas of recent construction. The speed limit data base is updated infrequently, with many areas with missing or incorrect speed limits - and because that data has remained incorrect for over a year, that data base is either based on a very poor information source or it isn't being updated very often. For offline routing, the NAV 1.0 software used to use traffic data that was downloaded every few minutes - unclear if current NAV software is also getting access to this slightly out-of-data traffic information for use when cloud server navigation is disabled. The onboard data is likely being used for display of the dashboard navigation map and for use when doing onboard navigation.
In addition to navigation, it also appears the onboard map data base is also being used for NOAP - which could be a major problem, because that map data is usually 1-2 years out of date, causing NOAP to have problems when highways have gone under recent construction or when exit/interchange ramps have been opened, closed or moved.
To get FSD working, Tesla can't rely on out-of-date map and speed limit data. The current strategy of relying on the onboard maps and speed limit data won't work. When the AP2 software starts reading traffic signs, that will provide more up-to-date information than the speed limit database. But the map data inaccuracies are harder to overcome.
What Tesla should do is move away from the "old school" strategy of downloading all of the map & speed limit data for huge regions (continents). The vast majority of this data will never be used - and only a very small amount will be used for a single driving session. What Tesla could do - when a route is calculated, the server could send down the small amount of map/speed limit data from the up-to-date database on the server down to the vehicle - and use that for the current driving session.
Especially for vehicles connected to the Internet almost all of the time, it makes much more sense to send the data only when needed, rather than downloading massive amounts of data - that will never be used...