Yup, you get sent a license file and you can put that either in the install directory, My Documents, or use the 'enter license' button in the about screen. Then, the free edition automatically upgrades to licensed - there isn't a different download.
@Fimbul: we suggested subscriptions earlier this year. We suggested having a small yearly fee and then you get Construct 3, Construct 4 etc. included with that. However, it was really unpopular and put off a large number of people. So we've gone for the traditional model of buy a major version then have the option of buying again when a new major version comes out.