The difference, as I understand it, is that the REPL is the running Lisp, and all of its code. You can inspect or redefine any function inside of Lisp itself, live, no matter how integral, in the same way you input any of your own code.
The interactive mode of interpreted languages is more of a sandbox, from that perspective.
The REPL often is also the default debugger. On error one gets a debug REPL at the point of the error (no unwinding), one debug level deeper. The debug REPL is basically a normal REPL, but with some debug features (stack movement, ...) and the ability to call predefined restarts in the code.
The interactive mode of interpreted languages is more of a sandbox, from that perspective.