Normally you would use something like the 'SelectedIndexChanged' or 'SelectedValueChanged' action of a listbox, otherwise events like DoubleClick will fail if the keyboard is used to change the selection.
Also that event should be triggered on the second listbox when it is changed by the first.
EDIT:
On a standard Listbox, there is no .SelectedItem.Value, only .SelectedItem. However as a listbox holds objects, not just text, you need to say you want the text value.
presentlyPlayingTextBox.Text = playlistListBox.SelectedItem.ToString();