With Passive Mode the Flic button will disconnect the bluetooth LE link 10 seconds after each press in order to save battery power. The downside of this is that the link has to be established again the next time you press the button so the click latency will be worse.
With Active Mode the Flic button stays connected for as long as possible while the phone is within range and will automatically reconnect once the phone comes into range. The idea of this is that the button should be ready to send the button events immediately when you press it. This may sound wasteful, but in reality this mode will still allow for a long battery life due to the conservative connection parameters we use.
So if you experience poor battery life on your Flic then you can try out Passive Mode. Keep in mind that either of these modes will not affect the Phone’s battery consumption. Also, I should mention that the setting is only applied on the next disconnect of the Flic (the next time you go out of range) since the connection parameters are negotiated during the connection phase.
I know that this information would have been hard to figure out on your own since it is not really documented anywhere.