Ok, so there's no clear distinction between both then?
Hey, thanks for the info GavinT !
Hmmm. How can I explain this??!!
OK. In the Tokyo area, there is the 'Metro' rail company, which runs the underground/metro/tube/whatever-you-want-to-call-the-going-underground-trains 
There are also trains run by JR (Japan Railways), which run overground. The main one in Tokyo is the Yamanote line, which is kinda like the 'Circle' line on the London tube, except it's not underground.
Then, you also get some other private train companies, who run lines that extend out from central Tokyo districts into the suburbs and surrounding towns (actually all the towns pretty much have merge together to make Tokyo this huge urban metropolis, with very little countryside
). Examples of such companies would be Keio, Keisei and Odakyu.
Now, this is the complicated bit. There are a couple of Metro lines, which have been build recently, such as the Oedo line, that are what would be considered 'British' style tube trains. They only run on the underground network, have small carriages, with curved roofs to save space.
The rest, they use standard trains, all the same shape square boxes. All the same width track. All overhead electrified wires.
(The bullet train, 'Shinkansen', however, uses a seperate rail system, with a different width track)
For example, it is possible for me to catch a train from my local station (Matsudo), to get to a place West of Tokyo called Karikada. Doing so, means I travel on 3(!!!) different rail networks, without changing train!
From Matsudo to Ayase, I am travelling on the (Local, not Rapid) line for the JR Joban line. At Ayase, the JR driver gets off the train, and the Metro driver gets on the train. At this point, the train just keeps going now, but it is now a Chiyoda Line train on the Tokyo Metro (it now goes undergound). This is the train I use to get to work, as I would get off at Otemachi (on the Metro) line. However, if I keep going I eventually reach Yoyogi-Uehara. There, the Metro driver gets off the train, and the Odakyu driver gets on the train. Now, the train has become a 'Rapid Express' on the Odakyu Tama-Express line!
Its nuts!! The fun bit is paying for all this, because you've travelled on 3 different networks without getting off the damn train. The ticket machines get quite complicated sometimes at some stations, because you'll have multiple companies sharing stations.
I know it sounds all complicated, but to be fair, they all run perfectly about 99% of the time.