The Internet is more than a single organism, though; it acts more like a cooperative neural network. When I link my computer to my network, my computer gets access to more information than it had previously. When I link my network to your network, my network (and each individual computer thereon) gets more access to information than it had before.
The super-organism known as the Internet is very smart, though... it has a bunch of symbionts taking care of it, making certain its nervous system stays intact (eventually through the movement of small pieces of paper printed with obscure patterns, which has become easy to facilitate through the movement of data carried on the Internet itself). The humans are in a symbiotic relationship to the Internet -- the Internet carries the data, and the humans take care of it. (And isn't it amazing how much of the traffic on the Internet's nervous system regards taking care of the Internet?)