bind-html-compile is not a standard Angular directive, it comes with the module https://github.com/incuna/angular-bind-html-compile and it is used to compile binded data.To make it simple, it is equivalent to write html in your source code: it will be re-evaluated and if other directives are found, they will work as expected.
ng-bind-html is a standard directive (bundled with Angular itself) and just output html strings without compiling it.
for example, if you controller has a variable with plain html, like in:
$scope.dataToDisplay = '<h1><strong>Title</strong></h1>';
Then you can go with
If you need to inject variables that contain html with other directives, such as:
$scope.dataToDisplay = '<h1 ng-show="showIfOtherVariable"><strong>Title</strong></h1>';
then you need to aforementioned module.