Delete Developer beta profiles
First, you have to delete the Developer Beta Software Profile that you’ve downloaded from the Apple Developer Center or from BetaProfiles.dev.
Go to Settings - General - VPN & Device Management and tap the Beta Software Profile that appears. Tap Remove Profile. If asked, enter your device passcode, then tap Remove. Then restart your devices for the changes to take effect.
Or you can skip this step because this will automatically delete the developer beta profile if you install the public beta profile in the next step.
Download Public beta profiles
From your device, go to BetaProfiles.dev or the Apple's Beta Software website (Requires sign in with your Apple ID) to download the public beta profile.
Your device will notify you when the profile has been downloaded. Open Settings, tap the profile and follow the onscreen instructions to allow installation.
That's it! And the next time there is an update to the public beta you should get the new build under the public beta portal.