A question about a recipe: Fresh Sriracha (aka, home made 'Rooster')

The biggest difference will be acidity level. Distilled white vinegar is 5% and it will be listed on the bottle the percent acidity. Since apple cider vinegar is usually more of a raw product, acidity levels vary considerably. Check the package of the vinegar you're using--it should be listed in the lower left hand corner.
The other reason is flavor. Apple cider vinegar has a distinctly sweet flavor, while white vinegar is very neutral. In a lot of instances, that sweetness from apple cider vinegar enhances a recipe.

My guess is that white vinegar was the choice because of both these reasons- consistency and neutrality. Consistent acidity will give the fresh sriracha substantially longer shelf life and will be a safer condiment in terms of not harboring bacteria. If you have a bottle of apple cider vinegar that has acidity level at 5% or above, you should try it and see how the flavor compares.