As I understand it Popcorn Time torrents and seeds a file sequentially, allowing you to watch the video as it downloads and removes the video from your device when it's over. It's more similar to torrenting a file yourself, but doesn't store the file when it's over. In contrast this torrents the file to the server and allows you to stream the content over your browser, so it isn't your device doing the leeching and seeding. You just watch it all happen.
Last I checked, it's actually a simple check-box in the settings to stop popcorn time from deleting the file afterwards. Really the only thing popcorn time did over the old-fashioned torrenting process was enable easy 'streaming' of torrents by downloading the bits in order instead of by least available first, as well as a nice UI to find them
