This is quite easy, actually. Television in the old days is simply a CRT modified to show images from radio signals, and it's main purpose (aside from government 'propaganda' in its various forms) is to entertain and show motion pictures inside the house.
The purpose of a computer, on the other hand, is to compute, i.e. to calculate complex algorithms and formulae. These devices were mostly housed in universities where math and physics professors actually have to book time to use one. Eventually, these profs want to share their data to profs in other universities, so they had to find a way for these computers to communicate with each other. Hence, the Internet.
Now, the Internet without television may look somewhat different from OTL's Internet circa 2009. Perhaps, there would be a focus on news articles, e-mail, blogs, games, and social networking (arising and spinning off from its original purpose), but most of the entertainment aspects of it such as videos and music may be a bit muted. In other words, it may look like the Internet at around 1997.