One of the arguments I saw for faster internet outside of a single-user app is when you have multiple people using a connection. If you have a family of 5 people all using a Cable connection in the US today, it is very easy to overload that line. And since upstream sucks on most home connections today (1.5 to 2 Mbps upstream), a single person uploading a video to youtube (or backing up data) will kill the connection for every other person using that line.
In the current ecosystem, I see gigabit connections being useful for multi-user connections where a lot of video is being streamed. Or using cloud storage solutions like Dropbox or Google Drive.
In the future, I could see other use cases. For what we know about Google's self driving cars, the first time a road is driven on (manual driving) it gathers a ton of data about that road. When you park in the garage, it may have to upload gigabytes worth of data for Google to process to make available to other cars. Doing that over the current connections most people have would be painful. If you have a large pipe for uploading lots of data, this becomes a lot more feasible.