There is no difference between a DSL router and a Cable router. They are both high speed connections and use Category 5 cabling. You may use any router you find.

If you are talking about a modem, then there is a difference. A cable modem will have a jack on it to connect to the cable TV lines. A DSL modem will have a jack on it to connect it to the phone lines.

Here is how it should be set up:

DSL Modem –> High Speed Router –> Computer
or if you have cable:
Cable Modem –> High Speed Router –> Computer

And in answer to your question about ADSL and DSL. ADSL is DSL but not the other way around. Kind of like a square is a rectangle but a rectangle is not a square. You would use a ADSL modem on your phone lines to get high speed internet though. No difference in routers.