Oracle Instant Client is a light-weight software which helps to make connection to Oracle Database. It contains OCI, JDBC, SQL*PLUS and SDK. In other hand the Oracle Call Interface (OCI) is an application programming interface (API) that allows applications written in C to interact with one or more Oracle Servers.

For details:
What is Oracle's instant client and how do I install it?
Introduction to Oracle Call Interface

Oracle Instant Client is a light-weight software which helps to run OCI, OBDBC, Pro*C, JDBC without installing full Oracle client. In other hand the Oracle Call Interface (OCI) is an application programming interface (API) that allows applications written in C to interact with one or more Oracle Servers.

For details:
What is Oracle's instant client and how do I install it?
Introduction to Oracle Call Interface