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.