My guess would be that when the website gets the Post and logs the user in, it sets cookies on the response to indicate that the user is logged in, and then requires those cookies on subsequent Get's.
You will need to do something like the following (this is borrowed from a bigger app so may not compile right out of the box)
DefaultHttpClient mHttpClient = new DefaultHttpClient();
BasicHttpContext mHttpContext = new BasicHttpContext();
CookieStore mCookieStore = new BasicCookieStore();
mHttpContext.setAttribute(ClientContext.COOKIE_STORE, mCookieStore);
This sets up a cookie store within the HTTP context, and you then use that context on Get's and Post's. For example...
HttpResponse response = mHttpClient.execute(mRequest, mHttpContext);
Under the covers the HTTP client will store cookies from responses, and add them to requests.