The Client SDK consists of a client and admin api. The admin api mainly dealt with operational aspects such as business network install/start/upgrade and card administration. The fabric node-sdk package fabric-client provides the capability to form hyperledger fabric operational aspects such as install and instantiate (which was what a network start did as well as some other aspects).
The client package was meant for general client side applications to interact with the chaincode/smart contract. Most of which has been covered by other sections. The final api, which is probably the most important one was submitTransaction.
The normal situation in Composer was that a transaction was something that resulted in a change to the world state and a block containing this change would be added to the blockchain. Composer used a submit/notify model so that the call returned a promise which was fulfilled when the peers had committed the transaction to the blockchain. the fabric-network api provides an equivalent capability and you would either use submitTransaction on the contract instance or submit on a Transaction instance.
In Composer you could also optionally declare if the transaction should return modelled data using the @returns() annotation if successfully committed to the blockchain. When you submit a transaction using the fabric network you can also get back returned data.
A feature of Composer was to define a transaction as a read only transaction, ie it should not commit it to the blockchain and optionally it could return a value as well (again using the @returns() annotation). In this case you would annotate the transaction with the @commit(false). To do the equivalent using the fabric-network api you should call evaluateTransaction on the contract instance or evaluate on a transaction instance and the result will be returned.