I presume your Outlook that works is on a PC which is _wired_ to the router, but no longer has an analogue phone-line connection ?
I.e. it could still be emailing via your PC's internal dial-up modem.
Control Panel> Internet Options> Connections> Never Dial a Connection
should be set on both your PC and laptop.
Unless you will be using dial-up on your laptop when you are away?