As a developer, I'm always comparing versions of files from here and there. As someone who collaborates on HTML, it's easy for people to step on each other when your only connection is a remote FTP server. BC let's you open up local and remote versions of files simultaneously and do a line by line comparison. Out of the box it's extremely powerful and intuitive, and with deep customization features it only gets better.
As a general computer user, I use BC all the time to compare folder structures, do quick backups to remote drives, stuff like that.
If you compare files and folders, or you should and you just cut and paste instead like I used to, you have to get this thing.