Just to add a bit to the discussion from my experience working with Volley:
Volley does not handle streaming uploads or downloads in any sense. That is, the entire request body has to be in memory and you cannot use an OutputStream to write the request body to the underlying socket, nor can you use an InputStream to read the response body, as basic HttpURLConnection does. So, Volley is a poor choice for uploading or downloading large files. Your requests and responses should be small. This is one of the biggest limitations of Volley that I have personally encountered. For what it's worth, OkHttp does have interfaces for working with streams.
The lack of official documentation is annoying, although I have been able to work around that by reading the source code, which is pretty easy to follow. What is more bothersome is that, as far as I can tell, Volley has no official release versions and no Maven or Gradle artifact, and therefore managing it as a dependency becomes more of a headache than, say, any of the libraries Square has released. You just clone a repo, build a jar, and you're on your own. Looking for a bug fix? Fetch and hope it's there. You might get some other stuff, too; it won't be documented. In my opinion, this effectively means that Volley is an unsupported 3rd party library, even though the code base is reasonably active. Caveat emptor.
As a nit, having the Content-Type tied to the class/request type (JsonObjectRequest, ImageRequest, etc.) is kind of awkward and reduces the flexibility of the calling code a bit, as you are tied to Volley's existing Request type hierarchy. I like the straightforwardness of just setting Content-Type as a header like any other (don't do this with Volley, by the way; you'll end up with two Content-Type headers!). That's just my personal opinion, though, and it can be worked around.
That is not to say that Volley does not have some useful features. It certainly does. Easily customizable retry policies, transparent caching, a cancellation API, and support for request scheduling and concurrent connections are great features. Just know that it's not intended for all HTTP use cases (see item 1 above), and that there are some headaches involved in putting Volley into production use in your app (item 2).