It's tempting to imagine self-publishing as a new expression for the old-fashioned term vanity publishing, however, there's actually a distinction between both.
A vanity publisher is an author who privately pays for bound and printed copies of their work for distribution and sale to a limited market. A vanity publisher will charge the author upfront for the entire print run, which is a set number, perhaps 500 copies. Vanity describes the motive behind an author following this course of action, and vanity publishing used to carry the stigma of the book not being good enough to secure the backing of a literary agent or publishing house.
Self-publishing is an expression that describes the method that an author follows to produce either a digital or printed book which is later accessible to a large population. It doesn't require an exclusive publisher however, it can be done by the author by himself. Book self publishing or indie publishing carries no stigma, with even some highly successful authors ditching their publishers and choosing to go it alone.
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Which is the best?
When a vanity book is published the author is not likely to receive any other benefit in exchange for the bound and printed copies of their book. A vanity press typically will not provide any assistance in marketing or distribution for the writer. The job of a vanity press is to produce the product. That's a different aspect.
While a print run of 500 might seem a huge amount to the author, it is actually quite a small print run compared to what a publishing house might commission, so the cost of those 500 books is rather high. Self-publishing allows authors to create a printed book or digital version for no cost if they opt to complete every step of the process themselves and publish through a business. With these platforms, authors can format their work as well as design a cover and make their book available to anyone around the world without spending any money.