I find the primary advantage of a REPL being able to "explore" the language, or code. If I am trying to figure out how to slice a particular list of items and then split the items on a regex into key-value pairs for a map, sometimes I just prefer instant feedback and being able to mess around with live variables. That being said I often find that after about 5 minutes in (Python's) REPL I start wishing I had just written a script and run it in the interpreter instead, as going back to edit functions or build classes can be painful. Stuff like iPython offer solutions to this problem by integration with editors and more.

The speed you get feedback is the key imo. When I use a language that has a repl I usually find myself using it more than the text editor / ide. And in my experience there's a nice side effect too: since you cannot write too much in the repl, you end up writing smaller functions that leads to more modular, reusable code.