They are not the same, clearly. One is used solely in the controller; the other is a directive on an input element.
But even in their application they differ.
When you use $watch the watched expression will be evaluated on every digest cycle, and if there is a change, the handler is invoked.
With ng-change, the handler is invoked explicitly in response to an event.
With $watch, change can come from anywhere: user action, controller function, service – all will trigger the handler.
With ng-change, the change is restricted to a user action on a particular input element.
It is worth to note also that ng-change works only in combination with ng-model – in other words, the ng-change expression is evaluated only when ngModel.$viewValue (refer to ngModelController documentation for more info) is changed, which typically happens in response to a user-initiated event.