In answer to GG
My job (among other computer hassles) partly involves resolving problems on these on a daily basis. For several years I also went out to customers sites on behalf of BT on contract, both installing and diagnosing broadband installations and faults.
I can assure you that the training manuals of BT, TalkTalk and several other BB suppliers all emphasise the need to keep the router constantly turned on, and from personal experience (on a large number of installations) I can say that it DOES make a difference
GG may be lucky in his location and have a good connection to his exchange. Most are not so lucky and need every scrap of extra speed that can be wormed out of BT's creaking local wires
As for the other points
"refresh the IP address" - why bother? Keeping a constant IP address helps remote access when you want to dial in to your network.. OK, you can use DynDNS, but even then the more constant the address the better
"minor corruptions in memory are removed". Huh?? Joke of the day surely? Routers are embdedded systems running Linux or a BSD variant. You don't get windows type memory issues. Never known it to be an issue. Typically I find routers will run for months without needing to be restarted. What occasionally happens is they will "hang" after someone / something tries a port scan attack or similar. Then they'll lock up and need restarting, but that can be very very rare. No need to forgo the benefits of leaving them on continuously just for that odd chance