EIM is subset of EAI. It is one approach we will use while integrating two or more applications data. usually we will go for EIM when there is large amount of data to be exchanged between and time gap between the Loads could possibly more.
EAI is at a larger level is sharing or exchanging the data between two more legacy systems. It could be immediate data exchange using message exchange or bulk imports and exports (EIM).
In Siebel perspective we use Siebel EAI as the term to specifically define data exchange or some other concepts for data exchange and data accessing with other systems.
EIM: Enterprise Integration Manager is the server component which Transfers the bulk data from one database to another database. We mostly use EIM to IMPORT DELETE MERGE UPDATE the data.
EAI: Enterprise Integration Application is also a server component which transfers the data at runtime. It will integrate two different Applications.