I'll give you the nickel tour.
Modems, in today's venacular are designed to support a connection for one device (computer) and one device(computer) only. Routers, because of their greater intelligence, are designed to support more than one device (computer).
I never heard of anyone having an "internal" ADSL modem before, but not to say it doesn't exsist, I just haven't heard of it. The reason why is ADSL needs a modem that acts as a "spliter", that is the ability to spilt an analog phone call from a digital "data" signal. That is what allows people with DSL to use their phone and the internet at the same time. Phones only take up a fraction of the bandwidth over copper phone wire (about 4 khz) which means there is plenty of room, almost 90%, of bandwidth left over to transmit data. All you need is a DSL modem to know how to divide out the analog from the data and Voila ADSL.
Routers are intelligent devices able to work not only with several devices on "A" network, but several devices on "different" networks. That is to say if all your equipment is using the IP address on the network of 192.x.x.x, and your ISP is using IP address of say 65.x.x.x. Your router is able to "route" information between 192 and 65. A modem cannot do that.
A modem will dial into, in the above example, of network 65.x.x.x and give the connected device a network id of 65.x.x.y to put your computer on that network.
That means that the router is intelligent enough to connect multiple devices (computers) to a network, but a modem is not.
Other "routers" will perform the same funtion! That is their creation by design.
Modems will not.
Please feel free to elaborate on your question if additional clarification is needed.
D