Me uses the current object instance.
MyClass uses the memberdefinition in the class where the member is called
Class Person
    Public Overridable Sub DoSomething()
        Console.WriteLine("Person")
    End Sub
    Public Sub useMe()
        Me.DoSomething()
    End Sub
    Public Sub useMyClass()
        MyClass.DoSomething()
    End Sub
End Class
Class Customer
    Inherits Person
    Public Overrides Sub DoSomething()
        Console.WriteLine("Customer")
    End Sub
End Class
Example Usage:
Dim c As New Customer
c.useMe()
c.useMyClass()
Output:
Customer
Person