Judging by the tags you've used on this question, it looks like you're getting confused.
CookieContainer and CookieCollection are used with HttpWebRequest which is when your program is a HTTP client. CookieCollection stores cookies associated with a single domain name. CookieContainer stored all the cookies over all the domain names.
In ASP.NET, HttpRequest.Cookies is HttpCookieCollection and is used when your program is a HTTP server. It stores the cookies sent by the client to the server. Because there is only one domain name (i.e. yours) there is no need for a 2-dimensional collection.
The two sets of classes (CookieContainer and CookieCollection vs HttpCookieCollection) are totally unrelated with each other.
In this post I've provided some basic background on the cookie collection classes in .NET, but I don't understand your question. What is it you're trying to do?