May I use apple cider vinegar in place of white distilled vinegar when making the fresh sriracha? What's the difference in using one over the other?

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.