This option has been around for a long time, right?
When distributing it indicates:
Private – Available as a dedicated app on Apple Business Manager or Apple School Manager
Then you can distribute your app via a unique link and matching redemption code. These redemption codes are only valid once and you can order a number of them via Apple (the quantity does not matter, otherwise it is free). Users are then sent to the App Store via this link, where they can install the app.
An Enterprise account works differently, as you have to make sure that you are hosting the app so that users download the app via this link without going through the App Store.
So for enterprise app, you have to host the app yourself, you have to arrange an update mechanism by yourself, the user has to approve a certificate from your organization and the apps will last for one year max, so it is wise to get a new version before the end of the year.
Private custom apps can only be created via a normal (non-enterprise) account and have the advantage of installing and updating via the App Store, in fact it does not have all the above-mentioned disadvantages of an enterprise app. However, the big advantage of the Enterprise app is that you can publish it without giving Apple’s opinion on it because you don’t have to be approved by Apple (well, in some cases Apple will revoke your enterprise license, then you have to make it pissed off, Facebook had that).
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