If your app still says HBO Max, you'll need to download the updated Max app, but you won't have to pay for a new subscription.
The company says that on launch day, the HBO Max app will automatically update to Max for a "large portion" of users. Some, though, will be prompted to download the new Max app when they open HBO Max.
If you have HBO through a cable/satellite/internet ... All you need to do is download the Max app on a supported device (more on those later), and simply sign in through your provider.