Self-publishing is when writers bear the cost of publishing and use self-publishing companies only (for ex. Amazon self-publishing) to publish the book. The writer does all the work of completing the book and making it publish-ready by writing, editing, designing a book cover, and everything else. Then, the writer takes the book to the self-publisher and bears the cost of publishing the book too.
Indie publishing, on the other hand, is a concept where a small (much smaller than a traditional publishing house) company takes up the onus of finding great content. The Indie-company then works in tandem with the writer to help him complete his or her book by offering various services including editing/proofreading, best book cover design, etc. the Indie publishing company bears the responsibility of printing, marketing, and selling the book.
Normally, Indie publishers seek out writers as against traditional publishing houses that writers go seeking.
Amazon Kindle self-publishing means doing all the work yourself; the design, the layout, the cover, the interior file, the uploading, the marketing, the advertising, the editing, etc.
Indie-publishers manageyour work. You might write the manuscript, but hire book-cover maker, out-source the editing, beta reading, interior design, etc. to any number of people in the indie publishing community or to a single company who does it all under the same umbrella, such as White Magic Studios.
As against the general misconception that indie publishing companies usually work on a limited number of books;publishers such as White Magic Studios, publish a great number of great books every year which is ever-increasing. They work on all genres; be it romance or horror, fiction or non-fiction, novels or educative books...
Indie-publishing focuses on selective but quality writing. So, you, as an author, would want to write some compelling work and outsource the rest to professionals who know the business.
These are basically just different words for the same thing; there's no “official” difference.
