I have gone ahead and followed the tutorial to which you referred: Call an Azure API app from a web app client authenticated by Azure Active Directory

1. Create an Azure API Api that returns contact data
2. Deploy the API App to Azure App Service.
3. Secure the API App using Azure Active Directory.

Then I was able to retrieve the token, and as you can see from the following demo, my code is no different than your code, except that it using a later version of the using Microsoft.IdentityModel.Clients.ActiveDirectory libraries that use Async.

Get access token from AAD

class Program{    static void Main(string[] args)    {        var authContext = new AuthenticationContext(Constants.AUTHORITY);        var credential =             new ClientCredential(Constants.CLIENT_ID, Constants.CLIENT_SECRET);        var result = (AuthenticationResult)authContext            .AcquireTokenAsync(Constants.API_ID_URL, credential)            .Result;        var token = result.AccessToken;        Console.WriteLine(token.ToString());        Console.ReadLine();    }}

Constants

AUTHORITY. The first segment of this is https://login.microsoftonline.com. The final segment is an allowed tenant. We set the allowed tenant at portal.azure.com, going to the Gateway for our application, and choosing Settings > Identity > Azure Active Directory > Allowed Tenants. My tenant is bigfontoutlook.onmicrosoft.com.

CLIENT_ID. We retrieve this client id from the application that we added to Azure Active Directory. Find this at manage.windowsazure.com > Active Directory > Your Directory > APPLICATIONS > Your Application > CONFIGURE. Once we have retrieved it, we must add it to our Gateway's Azure Active Directory settings in the Client ID field.

CLIENT_SECRET. We create/retrieve this in the same location that we retrieve our client id.

API_ID_URL. We retrieve this within the Gateway blade for our Web API App by choosing Settings > Identity > Azure Active Directory > App URL.

Here are the ones that work for me.

class Constants{    public const string AUTHORITY =     "https://login.microsoftonline.com/bigfontoutlook.onmicrosoft.com/";    public const string CLIENT_ID =       "0d7dce06-c3e3-441f-89a7-f828e210ff6d";    public const string CLIENT_SECRET =      "AtRMr+Rijrgod4b9Q34i/UILldyJ2VO6n2jswkcVNDs=";    public const string API_ID_URL =       "https://mvp201514929cfaaf694.azurewebsites.net/login/aad";}

Final decoded JWT

This is what the decoded JWT access token contains.

{ typ: "JWT", alg: "RS256", x5t: "MnC_VZcATfM5pOYiJHMba9goEKY", kid: "MnC_VZcATfM5pOYiJHMba9goEKY"}.{ aud: "https://mvp201514929cfc350148cfa5c9b24a7daaf694.azurewebsites.net/login/aad", iss: "https://sts.windows.net/0252f597-5d7e-4722-bafa-0b26f37dc14f/", iat: 1442346927, nbf: 1442346927, exp: 1442350827, ver: "1.0", tid: "0252f597-5d7e-4722-bafa-0b26f37dc14f", oid: "5a6f33eb-b622-4996-8a6a-600dce355389", sub: "5a6f33eb-b622-4996-8a6a-600dce355389", idp: "https://sts.windows.net/0252f597-5d7e-4722-bafa-0b26f37dc14f/", appid: "0d7dce06-c3e3-441f-89a7-f828e210ff6d", appidacr: "1"}.

Note: It's a throwaway app in a throwaway active directory account with a throwaway resource group, so showing my security credentials is a non-issue.

Diagram just to be sure :)