The main point of the paper maps is so that players can reference the geography of the board while having secret discussions away from the table. My group ended up making a few higher quality laminated maps for this purpose as the paper maps are a bit small. For a typical in-person game of Diplomacy, there is a "main room", where the board lives, and a number of auxiliary locations, where players can have secret discussions (other rooms, outside the house, hallways, etc.). Having a reference of the map while in an auxiliary location is key as remembering where units are is much less to keep in your head than the entire geography of the board in your head. There is the added benefit of aiding in secret discussions in auxiliary locations that don't have good sound insulation, as players can point to locations on the map rather than saying them. When people in my play group go off to an auxiliary location, they almost always bring their order sheets and a paper map with them.
The secondary point of the paper maps is so that players can have a reference for the map (for territory abbreviations and adjacency) that they can look at closely without giving away to other players where on the board they are looking.
While you can write on the paper maps, this isn't usually of very high utility since the state of the board changes every turn. If people in my play group do write on one, it's in pencil and to plan out and agree on a particularly complicated maneuver involving a discrete set of units (and they will probably erase it afterwards). People in my group keep track separately of where the relevant pieces are, such as writing down their units on their order sheet (which they will need to do anyways to submit orders) and/or taking a picture of the board with a phone (pre-phones, people would periodically pop back into the main room to look at the board to remember where units are). We also occasionally write on a paper map to save the game state if we're putting it down for a session.
If you have people that really like writing on paper maps, it is probably worth getting a few higher quality maps on stiff paper laminated, and getting white board markers to go with them so that they erase better (if you don't have a laminator, packing tape can work as an alternative). We have one such map that I keep with my diplomacy set.