Use short hand/automatic {get;set} in class instead of making the varibale public

Question

I'm fairly new to C# and I'm following this tutorial.

I've reached the part about get and set of private instances (using theirs example)

class Person
{
  private string name; // field
  public string Name   // property
  {
    get { return name; }
    set { name = value; }
  }
}

class Program
{
  static void Main(string[] args)
  {
    Person myObj = new Person();
    myObj.Name = "Liam";
    Console.WriteLine(myObj.Name);
  }
}

where I assume get is used when we just do myObj.Name and set is called when using myObj.Name = "something" (right?).

They then further write that the code above can be shortend by doing

class Person
{
  public string Name  // property
  { get; set; }
}

class Program
{
  static void Main(string[] args)
  {
    Person myObj = new Person();
    myObj.Name = "Liam";
    Console.WriteLine(myObj.Name);
  }
}

and this I don't get, since now we aren't modifying any private property (name in the first code) but we just have one public property called Name. If that is the case, that we just have public string Name, then why would we need the {get;set}, now that Name is public? Shouldn't we be able to get and set it anyway?

EDIT

This question is not about what the get/set method does but what problem the Name {get;set;} solves instead of just making Name public.

I'm not talking about if e.g get returns only the first 3 letters of the Name field, I'm talking about the litteral {get;set;} i.e when nothing more than {get;set;} is defined, and what problem that solves.

If Name {get;set;} allows the user to alter the value of the auto-created private field _Name when why even bother having the _Name field? To me it looks like, I have a piece of paper that no one can see (private _Name) but, you can just send me a mail asking for the content of the paper ( Person.Name), and how you want it to be changed (with no restrictions) (Person.Name="Liam"). Why wouldn't I just give you the piece of paper and let you read/write what you want (just make Name public)?

The same way here, if there's no restrictions to {get;set;} then intead of having those methods modifying the _Name field why not just make Name public and ommit the {get;set;} part?

I'm aware that you can modify the get or set to do something specific, but that is not the question here. It is all about only the public string Name {get;set;}.

So, in the code below (not if we tweak it, change it or anything, but the code as it stands) why would we use the get_setter_name instead of just public_name?

class Person
{
    public string public_name;
    public string get_setter_name
    {
        get; set;
    }

}
class Progam
{
    static void Main(string[] args)
    {
        Person person = new Person();
        person.public_name = "Foo";
        Console.WriteLine(person.public_name);
        person.get_setter_name = "Bar";
        Console.WriteLine(person.get_setter_name);

    }

}