private_download_url method generates a non-CDN URL that access Cloudinary using a signature based on the API Secret of your Cloudinary account for downloading an image of type 'private' (such images cannot be accessed through the CDN, only allowed derived resources of them can be accessed through the CDN). This is the approach described in the blog post: http://cloudinary.com/blog/how_to_quickly_build_a_stock_photo_site_using_cloudinary.
This is relevant for delivering the private images in a 'slow' non-CDN way to authorized users.
signed_download_url method is used only for images of the 'authenticated' type. Such images, both originals and derived, are available for download through a CDN with a special signature. This option is currently supported only using the Amazon CloudFront CDN (not supported using the Akamai CDN, which is currently Cloudinary's default CDN layer). It uses AWS private & public key-pair of your AWS account for generating a CDN signed URL. Required for displaying authenticated content to any site visitors.
signed_download_urlmethod is currently only implemented in the Ruby SDK.