Downloading pdf file using WebRequests

Skip the BinaryReader and BinaryWriter and just copy the input stream to the output FileStream. Briefly

var fileName = "output/" + date.ToString("yyyy-MM-dd") + ".pdf";
using (var stream = File.Create(fileName))
  resp.GetResponseStream().CopyTo(stream);

Why not use the WebClient class?

using (WebClient webClient = new WebClient())
{
    webClient.DownloadFile("url", "filePath");
}

Your question asks about WebClient but your code shows you using Raw HTTP Requests & Resposnses.

Why don't you actually use the System.Net.WebClient ?

using(System.Net.WebClient wc = new WebClient()) 
{
    wc.DownloadFile("http://www.site.com/file.pdf",  "C:\\Temp\\File.pdf");
}