ADSL's "dial-up" nature is a throw back to the fact BT rolled it on on horrible USB adapters.
ADSL routers and standalone modems are basically the same as cable modems (which are mini-routers in themselves just with NAT abilities turned off usually). These are "always on" (assuming you leave the thing switched on). Rebooting your PC doesn't require reconnection in this case.
There is authentication involved with ADSL with username/password and this must be set up in the router/modem, whereas cable is authenticated on MAC address (easily faked, but then you're sharing the data with all your neighbours so who cares about security
).