There are a number of key tradeoffs between a private (a.k.a. permissioned) blockchain network and the public, main Ethereum network. Here are just a few of them:
- private blockchains require specifically and individually identifying the participants in the network, either through group consensus or some sort of designated gatekeeper
- private blockchains can rely less on internal incentives and more on the ability to pursue legal action against specifically known and identified participants, both at the network and at the application level
- private blockchains are significantly cheaper to maintain, with no need to directly fund economic incentives that secure consensus (instead relying on identification and contractual obligations of participants to motivate performance of various roles)
- for the same reason private blockchains may not need to contain internal tokens valued by an open market and may be able to reduce regulatory and reporting obligations as a result
- if the participants in a private chain trust each other enough, private chains may be capable of maintaining higher levels of data privacy than public blockchains
- public blockchains provide significantly stronger evidence of events having occurred since it is more difficult for a small set of entities to manipulate the record of transactions and resulting state
- public blockchains are significantly more effective at being censorship resistant
- for many smaller applications, use of an existing public blockchain will provide more security per dollar spent compared to constructing and maintaining a private chain.
Of course, there is no need for a given use case to make a firm decision between the two approaches, since atomic cross-chain trading and sidechaining can be used to build applications which take advantage of both private and public blockchains in order to maximise their total value. There is also an extremely thin line between a private blockchain and a heavily engineered state channel based off a public one and utilising the same network of participants.
A more detailed discussion on private and public blockchains is available on the Ethereum blog.