Quick reply.
You don't need Internet on N900 to get GPS tracking and logging.

To see the map tiles, you need either to draw them by hand, to download them through ModRana (Internet on N900 required) or to download them through PC's Internet connection.

You don't need Internet to see places (cities, towns, villages).

You do need Internet to get routes (from Google); it must be caused by the fact the program doesn't know where the streets and roads are situated.

A feature request to MartinK: let user to draw a route manually (in absence of Internet), and then give him turn-by-turn voice navigation for this route.

You do need Internet to search (Google) for restaurants nearby.

I haven't figured out how to use Points Of Interest, though.