Ok folks, I finally got back to messing about with this old lappy.
Well, because their was a couple of Windows programs that I needed to check out, without risking them on my main machines, I shoved XP on this lappy and tested them on it. Then, after doing this, I had a think about it until I decided eventually to partition the hdd and shove Linux Mint 14 (Petra) with the Cinnamon desktop on the primary partition.
I put Mint on a 25gb partition and left XP on the 15gb secondary partition.
The Sitecom usb wi-fi adapter that worked with XP, I left plugged in. After installation I connected to the internet with a network cable, installed a bunch of updates for Mint then from the repositories downloaded and installed that Broadcom STA driver that I had some success with on the Dell lappy.
I then rebooted the lappy.
Bingo....the wireless networks in the area were showing. I clicked on my router's SSID and typed the password.......I'm in!
I also ran that lsusb command that B.Eagle suggested, there at the top was my Sitecom usb adaptor.
To be honest, I'm still not sure if it was the updates or that Broadcom driver that did the trick, whichever....it's running on rails.
Now then, this Mint 16 distro, I still can't see what the improvements are over the Mint 14 that I run on my netbook. Not sure that I prefer the Cinnamon desktop to the Mate one either. In fact I'm looking at changing the Cinnamon desktop to Gnome if that's possible.
Eventually, I'm gonna take a look at another distro, maybe Fedora.
I'm open to suggestions though!