The short answer to this is “not a lot”. For reasons such as distancing themselves from self-published claptrap (because some of it is), some authors decided to call themselves independent publishers. I am sure there is more history to it than that, but a self-published author publishes their work (with or without outsourcing). An Indie publisher is an independent publisher. These can be an individual publishing their own books (self-publisher), a small publishing business that publishes an author with multiple pen names, or a small selection of authors. Indie publishers can be big or small operations.
A self-publisher can call themselves an indie publisher. But an indie publisher would rarely refer to their business as self-publishing. The semantics of the name means that an indie publisher removes the stigma of being self-published. This is important to some people and also important in marketing your books. The general voracious reader won’t care who produced your book, but some locations, such as libraries and bookstores, are less likely to stock books by self-published authors.
Whichever way you decide to go, best of luck.