I don't know of any System.Net.WebBrowser, but WebClient is basically a class that lets you easily download files (including html pages) from the web into memory or even directly to file. A basic code sample looks like this:
string html;
using (var wc = new WebClient())
{
html = wc.DownloadString("http://stackoverflow.com/questions/1780679/");
}
You can do a lot with WebClient, but there are some limitations. If you need to do some serious web scraping, you'll need to get lower level. That's where the HttpWebRequest/HttpWebResponse come in. You can use them to send any request a normal web browser might send, in any sequence. For example, you may need to authenticate with a web site before you can request the page you really want, and WebClient might not be able to do that. HttpWebRequest will.
Now, there is one other option. System.Windows.Forms.WebBrowser is a control designed to place on a form. It basically wraps the engine used in Internet Explorer to provide all the capabilities of a web browser. You need to be careful using this for general scraping: it's not portable (bad for mono), uses a lot of resources, has similar security issues as running a full browser, and has side-effects such as potentially leaking popup windows. The control is best used in a form to connect to a specific known web resource. For example, you may have a Windows Forms app for sale, and web app where you sell it for download. You might provide a WebBrowser control that shows a few pages on this web site specifically intended for view in your app that allows users to purchase in-app upgrades.