When I should consider to use Ethereum Private network for Production use case?
Can I dictate network difficulty rules and apply permission on such network.
You should consider using a private chain if there are variables of the network that you want control over that a public chain would not give you. For example, on a private chain, you can adjust anything you want, including the difficulty, the block time, the consensus mechanism, etc.
The problem with private chains is that it is likely to be more beneficial and efficient to simply use a traditional database. They provide much higher speed that the blockchain simply cannot offer (at this time). Additionally, with a private chain, you may "change history" of the blockchain at any time, so people may question your integrity.
What you may be looking for is a consortium chain. A consortium chain is similar to a private chain, except there are a number n various entities that have power to achieve consensus. By structuring your blockchain in this way, you can trust that if one bad actor attempts to manipulate the system, other good actors will stop them. All of this can occur with a customized blockchain that is more efficient than the public chain.