What's the difference between Task.Start/Wait and Async/Await?

I may be missing something

You are.

what is the difference between doing Task.Wait and await task?

You order your lunch from the waiter at the restaurant. A moment after giving your order, a friend walks in and sits down next to you and starts a conversation. Now you have two choices. You can ignore your friend until the task is complete -- you can wait until your soup arrives and do nothing else while you are waiting. Or you can respond to your friend, and when your friend stops talking, the waiter will bring you your soup.

Task.Wait blocks until the task is complete -- you ignore your friend until the task is complete. await keeps processing messages in the message queue, and when the task is complete, it enqueues a message that says "pick up where you left off after that await". You talk to your friend, and when there is a break in the conversation the soup arrives.

To demonstrate Eric's answer here is some code:

public void ButtonClick(object sender, EventArgs e)
{
  Task t = new Task.Factory.StartNew(DoSomethingThatTakesTime);
  t.Wait();  
  //If you press Button2 now you won't see anything in the console 
  //until this task is complete and then the label will be updated!
  UpdateLabelToSayItsComplete();
}

public async void ButtonClick(object sender, EventArgs e)
{
  var result = Task.Factory.StartNew(DoSomethingThatTakesTime);
  await result;
  //If you press Button2 now you will see stuff in the console and 
  //when the long method returns it will update the label!
  UpdateLabelToSayItsComplete();
}

public void Button_2_Click(object sender, EventArgs e)
{
  Console.WriteLine("Button 2 Clicked");
}

private void DoSomethingThatTakesTime()
{
  Thread.Sleep(10000);
}

This example demonstrates the difference very clearly. With async/await the calling thread will not block and continue executing.

static void Main(string[] args)
{
    WriteOutput("Program Begin");
    // DoAsTask();
    DoAsAsync();
    WriteOutput("Program End");
    Console.ReadLine();
}

static void DoAsTask()
{
    WriteOutput("1 - Starting");
    var t = Task.Factory.StartNew<int>(DoSomethingThatTakesTime);
    WriteOutput("2 - Task started");
    t.Wait();
    WriteOutput("3 - Task completed with result: " + t.Result);
}

static async Task DoAsAsync()
{
    WriteOutput("1 - Starting");
    var t = Task.Factory.StartNew<int>(DoSomethingThatTakesTime);
    WriteOutput("2 - Task started");
    var result = await t;
    WriteOutput("3 - Task completed with result: " + result);
}

static int DoSomethingThatTakesTime()
{
    WriteOutput("A - Started something");
    Thread.Sleep(1000);
    WriteOutput("B - Completed something");
    return 123;
}

static void WriteOutput(string message)
{
    Console.WriteLine("[{0}] {1}", Thread.CurrentThread.ManagedThreadId, message);
}

DoAsTask Output:

[1] Program Begin
[1] 1 - Starting
[1] 2 - Task started
[3] A - Started something
[3] B - Completed something
[1] 3 - Task completed with result: 123
[1] Program End

DoAsAsync Output:

[1] Program Begin
[1] 1 - Starting
[1] 2 - Task started
[3] A - Started something
[1] Program End
[3] B - Completed something
[3] 3 - Task completed with result: 123

Update: Improved example by showing the thread ID in the output.

Wait(), will cause to run potentially async code in sync manner. await will not.

For example, you have an asp.net web application. UserA calls /getUser/1 endpoint. asp.net app pool will pick a thread from thread pool (Thread1) and, this thread will make a http call. If you do Wait(), this thread will be blocked until http call resolves. While it is waiting, if UserB calls /getUser/2, then, app pool will need to serve another thread (Thread2) to make http call again. You just created (Well, fetched from app pool actually) another thread for no reason, because you cannot use Thread1 it was blocked by Wait().

If you use await on Thread1, then, SyncContext will manage sync between Thread1 and http call. Simply, it will notify once http call is done. Meanwhile, if UserB calls /getUser/2, then, you will use Thread1 again to make http call, because it was released once await got hit. Then another request can use it, even further more. Once http call is done (user1 or user2), Thread1 can get the result and return to caller (client). Thread1 was used for multiple tasks.

In this example, not much, practically. If you are awaiting a Task that returns on a different thread (like a WCF call) or relinquishes control to the operating system (like File IO), await will use fewer system resources by not blocking a thread.

In the example above, you can use "TaskCreationOptions.HideScheduler", and greatly modify the "DoAsTask" method. The method itself is not asynchronous, as it happens with "DoAsAsync" because it returns a "Task" value and is marked as "async", making several combinations, this is how it gives me exactly the same as using "async / await":

static Task DoAsTask()
{
    WriteOutput("1 - Starting");
    var t = Task.Factory.StartNew<int>(DoSomethingThatTakesTime, TaskCreationOptions.HideScheduler); //<-- HideScheduler do the magic

    TaskCompletionSource<int> tsc = new TaskCompletionSource<int>();
    t.ContinueWith(tsk => tsc.TrySetResult(tsk.Result)); //<-- Set the result to the created Task

    WriteOutput("2 - Task started");

    tsc.Task.ContinueWith(tsk => WriteOutput("3 - Task completed with result: " + tsk.Result)); //<-- Complete the Task
    return tsc.Task;
}