In terms of functionality that is lost if you don't have enter your login:
- you won't obtain market and outfitting details. This is used for checking commodity purchase and sales, but also sends this data to EDDN to power systems such as https://eddb.io/
- you won't be able to export your ship to Coriolis
- some information on the current star system and station you are docked at will be unavailable
It's certainly a lot less information than it used to be with the advent and evolution of the game journal, and we're trying to convince FD to make it entirely redundant. There isn't a third party developer out there that doesn't want to see this data available either locally or with a token-based system, and it's totally understandable that you do not want to put your login details in to a random program. If you're technical then you can look at EDDI's source code (link in the OP) and compile it yourself to be sure.
Not sure exactly what you mean by "traffic comms" but EDDI should still tell you about docking pad when you're approaching a station, for example, regardless of if you're logged in to the companion API.