<p>I have read that if I disconnect OctoPrint when printing, the print will stop.  Since I thought the advantage of OctoPrint over, say, printing from Cura, was that it didn't tie up the computer while the print was taking place, what are the advantages of OctoPrint?</p>

The benefit of using OctoPrint as a printserver lies in the fact that it can be used on a stable computer platform. E.g. when you install OctoPrint on a Raspberry Pi, you are ensured that the "computer" stays online. Other platforms, such as Windows are much prone to interrupt the printing process (user actions during printing, sleep mode, Windows updates, etc.). Furthermore, the power consumption of the Raspberry Pi is also much lower than a full computer or laptop running OctoPrint.
An advantage of running OctoPrint on a dedicated Raspberry Pi is that you can access it from anywhere within your network through a browser interface (or even from the outside) as it is always online. Other advantages of OctoPrint over e.g. Cura is that there are numerous plugins available to tailor the printing process to your needs.