A TNC is just a modem for RF. It lets your computer use the radio as the transmission medium. Most of the digital modes that you can use with your computer don't need a TNC; they just interface with your soundcard. The digital modes let you chat with other people same protocols on their computers. It's just like online chat, only you're doing over the air. You can also send & receive pictures, files, etc.
The TNC is really only needed for certain modes or protocols, or where you want the radio to be able to handle digital traffic in the absence of a computer. For example, some modes like Pactor are called ARQ modes. That means that the error correcting is handled every few packets, with the receiving radio sending acknowledgments or retry requests very quickly throughout the transmission of the message. Right now there aren't very good ways to handle this using just a soundcard. That's where the TNC or RF Modem comes in.
With APRS, you need a TNC built in to the radio if you want to be able to read the messages on the radio screen and also be able to transmit the messages. The alternative is to use a computer and TNC attached to the radio. You don't really need a TNC for the computer to do this, but the configuration becomes a little cumbersome.