Does 織機 mean the device, while 機 means the concept?
Or maybe one of them is more formal than the other?
Both refer to the device of weaving, but modern automatic power-looms are usually called 織機 (しょっき) or 織布機 (しょくふき). Traditional, manual looms are typically called 機 (はた) or 機織り機 (はたおりき) in novels and fairy tales. In academic contexts they're called 織機 anyway (see Japanese Wikipedia on 織機).