In what situation should I use a Private Set on a property versus making it a ReadOnly property? Take into consideration the two very simplistic examples below.

First example:

Public Class Person

    Private _name As String

    Public Property Name As String
        Get
            Return _name
        End Get
        Private Set(ByVal value As String)
            _name = value
        End Set
    End Property

    Public Sub WorkOnName()

        Dim txtInfo As TextInfo = _
            Threading.Thread.CurrentThread.CurrentCulture.TextInfo

        Me.Name = txtInfo.ToTitleCase(Me.Name)

    End Sub

End Class

// ----------

public class Person
{
    private string _name;
    public string Name
    {
        get { return _name; }
        private set { _name = value; }
    }

    public void WorkOnName()
    {
        TextInfo txtInfo = System.Threading.Thread.CurrentThread.CurrentCulture.TextInfo;
        this.Name = txtInfo.ToTitleCase(this.Name);
    }
}

Second example:

Public Class AnotherPerson

    Private _name As String

    Public ReadOnly Property Name As String
        Get
            Return _name
        End Get
    End Property

    Public Sub WorkOnName()

        Dim txtInfo As TextInfo = _
            Threading.Thread.CurrentThread.CurrentCulture.TextInfo

        _name = txtInfo.ToTitleCase(_name)

    End Sub

End Class

// ---------------

public class AnotherPerson
{
    private string _name;
    public string Name
    {
        get { return _name; }
    }

    public void WorkOnName()
    {
        TextInfo txtInfo = System.Threading.Thread.CurrentThread.CurrentCulture.TextInfo;
        _name = txtInfo.ToTitleCase(_name);
    }
}

They both yield the same results. Is this a situation where there's no right and wrong, and it's just a matter of preference?