Voidhawk, Richard,
thanks for the suggestions. I've already tried both of them (I think):
plugging the notebook in to the same phone connection as my desktop - same problem.
It could be the lead from the phone socket to the modem, I'll try to get a replacement lead and see if that works. I can't try your suggestion, Richard, unplugging the lead to the desktop at the modem and connecting that to the laptop because that lead has a different plug to the lead to the laptop modem which is a PCMCIA card. (Desktop modem socket is a standard US phone connector, laptop modem socket is a flat plug.)
Re-installing W98 doesn't seem to make any difference (that was actually one of the first things I tried - several times! - and the repair place I took it to also tried that). Although I did wonder if the installation for a laptop differed in any significant way to that for a desktop - I seem to remember that there was an option like that during the installation procedure. I thought I'd got it right, maybe not!
It just seems very strange for two apparently identical modems both give trouble, makes me think that there is something inside the machine!