var
    adoSave:  TADOConnection;
begin
    ......................................
    ......................................
    if adoSave<>Nil then begin 
      ShowMessage('Not_Nil');
      adoSave.Close;       -- соответственно здесь ошибка
      FreeAndNil(adoSave);
    end;
end

procedure TForm1.Button4Click(Sender: TObject);
var s: TADOConnection;
begin
  if(S=Nil)then ShowMessage('Nil') else ShowMessage('Not Nil')
end;

type
  TForm1 = class(TForm)
   
    ADOConnection1: TADOConnection;
    procedure Button4Click(Sender: TObject);
  private
    { Private declarations }
  public
    { Public declarations }
  end;

var
  Form1: TForm1;

implementation

{$R *.dfm}

procedure TForm1.Button4Click(Sender: TObject);
var s: TADOConnection;
begin
  s:=Nil;
  if(ADOConnection1=Nil)then ShowMessage('Nil') else ShowMessage('Not Nil')
end;

end.