Modem is just Modulation Demodulation, the "sign in settings" for your ISP are here, this is converting an analogue signal form the ISP to a Digital signal for your router.
In the UK most Home internet other than virgin media and the new FTTC was provided with a spliter in the phone socket and a rj11 into your "router" into, this was an ADSL2+ modem + router in one box and usually PPPoA connection for the ISP account details
With Virgin media or any FTTC product they provide a modem during the installation, although now some of the VDSL isps (sky bt) have a router modem combo, it does your VDSL over rj11 again, though they can be used with an external modem too.
In theory you could have seperate boxes for Modem, Gateway, DHCP, Switch, Gigabit networking, Wireless networking... this was the case from some crazy setups.
We mostly are used to getting a router (Modem 4*ethernet + wireless) with our Broadband since 2006 as they are kinda required to make it work. In theory you never have to use their stuff BUT it may be in the contract that you do.
N600 is not limiting the search enough with my google fu, so I have found multiple models for ADSL and PPPoE so Cant advise on that.
Need to know what ISP set-up you have but in general on ADSL2+ (up to 24mbps) the ISP provided router has a modem that is going to be within 0.3snr of any other you get in my experience. If your going for fibre from anyone soon you can use a different box for the modem but the white one openreach provide is what I use with my Asus RT66U N router, works a treat.
I then got FTTP and that has a ONT (optical networktermination) that plugs into a fibre "modem" not sure what it really is called, late to go downstairs and check. Sufficed to say I checked the settings (same as using a VDSL modem, PPPoE with bthomehub default user name with no password)
P.S. It's late and I have used this as an excuse to waffle, I know somewhere I said modem is where ISP sign in stats go, thats wrong, but I will format later
