Think of it like the NTP protocol to for machines to sync wall-time. The idea is that there's an authoritative source of data, in this case the local time on the server. Your machine also does its best to independently track and maintain the current time. But it may be fast or slow, or experience some other issue that causes it to drift. And the longer its left by itself without syncing to an NTP server, the larger that potential drift.
So the server and the client are both independently calculating the state. But whatever the server calculates is authoritative, and the client's version is just a local simulation that is very accurate, but not 100%. Especially when it comes to other players, where your client is typically just executing under simplistic assumptions of things like constant forward movement.
However, luckily for the client, it receives a data sync from the authoritative server several times a second, so all that potential drift gets quickly corrected while the differences are still extremely small. For your own actions, where your client had access to all your input, the differences are probably even too small to notice. For other players though, it will be more noticeable, especially if there's an abrupt change in input.