It does make quite a big difference. The network packets received from the server in Quake will tell you exactly what state the game is in at any point in time. They contain information about the position and state of every entity and their motion, compressed via delta encoding. That means there's very little room for misinterpretation on the client side that would lead to de-sync issues. In fact clients have quite a lot of freedom in how they want to represent said game state, and can for example add animation interpolation to smoothen things out.

The example you mention of demo playback de-syncing when the circumstances slightly change, that is exactly what you get when you only record inputs from the player. Doom actually did this too for its networking model and demo playback system. That relies much more on the engine being deterministic and the runtime environment behaving consistently, because each client that replays those inputs has to run the exact same game simulation, in order for the resulting game states to match.