[madman]

Esta funcionandoooooooo....!!!

 
uses Wininet;

function GetInetFile
(const fileURL, FileName: String): boolean;
const BufferSize = 1024;
var
  hSession, hURL: HInternet;
  Buffer: array[1..BufferSize] of Byte;
  BufferLen: DWORD;
  f: File;
  sAppName: string;
begin
 Result:=False;
 sAppName := ExtractFileName(Application.ExeName);
 hSession := InternetOpen(PChar(sAppName),
                INTERNET_OPEN_TYPE_PRECONFIG,
               nil, nil, 0);
 try
  hURL := InternetOpenURL(hSession,
            PChar(fileURL),
            nil,0,INTERNET_FLAG_RELOAD,0); // Aqui es donde modifique... puse INTERNET_FLAG_RELOAD.

  try
   AssignFile(f, FileName);
   Rewrite(f,1);
   repeat
    InternetReadFile(hURL, @Buffer,
                     SizeOf(Buffer), BufferLen);
    BlockWrite(f, Buffer, BufferLen)
   until BufferLen = 0;
   CloseFile(f);
   Result:=True;
  finally
   InternetCloseHandle(hURL)
  end
 finally
  InternetCloseHandle(hSession)
 end
end;

Muchas gracias