In its oldest English sense, dating from the 15th century,1 a repository is a place where things are stored, usually for safe keeping. Depository, which entered English a couple of centuries later,2 bears the same meaning (though, considered etymologically, a depository is a place where things are deposited, whereas a repository is one where things rest), and indeed both words are widely used to refer to places where things are stored.
In uses outside that general sense, though, there are some subtle differences in how the words are used. Repository is used more broadly and as a result is about three times as common. In this century, at least, depository is largely confined to financial contexts—especially in reference to institutions that store money and other assets—but this is by no means a rule, and it does sometimes appear where repository is the usual word. Repositories, meanwhile, tend to be places where art, documents, artifacts, and other nonmonetary treasures are stored. It’s also more commonly used in figurative senses; for example, a very smart person is a repository of knowledge, and a troubled city might be a repository of discontent.
Also, depository doubles as an adjective meaning of or relating to deposits. A depository receipt, for instance, is a written acknowledgment of a deposit.
A third word, depositary, is also often mixed up with depository. They are considered variants of each other in legal contexts especially,3 but depositary usually denotes people (with whom things are deposited), while depository usually denotes places.
This allowed investment companies to avoid the restrictions imposed on depository institutions, while offering bank-like products stemming from the product innovations. [Bank Management, Timothy W. Koch and S. Scott MacDonald]
Mr. Michel and another researcher, Erez Lieberman Aiden, led a project to mine the virtual book depository known as Google Books and to track the use of words over time. [New York Times]
The depository will be large enough to conserve a copy of all distinct accessions now held in genebanks around the world. [The State of the World's Animal Genetic Resources for Food and Agriculture, Barbara Rischkowsky]
Search histories are a depository of our deepest fears. [Guardian]
But the Coliseum became a repository for all the team’s troubles, a convenient scapegoat. [Newsday]
The policies should stipulate that when authors publish in non-OA journals, they should retain their rights to self-archive in their institutional repositories. [Qualitative and Quantitative Methods in Libraries, Anthi Katsirikou and Christos H. Skiadas]
Is your kitchen drawer a repository of half-empty matchboxes, bits of string, rubber bands, jar lids, screws, dried-up ballpoint pens, corks and bottle caps, unidentified keys, scraps of Christmas paper and those little scribble pads that real estate agents send you? [Nelson Mail]
[H]e is a repository of all those yearnings after the Eternal Vision which are expressed in the sigh of the lover or the groan of the martyr [Blake's Humanism, John Beer]