Those customers that got "badly burned" are going to want to know that you've learned your lesson.
If the event hit the press or word got around to your target customer base, you'll need to convince them that it won't happen again (I'm looking at you, Southwest Airlines).
If your industry is one where the failure could cause death or injury if it happened again - even to a competitor - then you have a moral and possibly legal obligation to "go public" within your industry so they can learn from your experience (I'm looking at you, Blue Bell Creamery).
Even if it's not life-or-death, you may find it good busine$$/good PR to share details within your industry or to the general public (thank you, Google).
There are some cases where publicity isn't critical.
For example, if you sell widgets and you had a no-critical-lessons-learned systemic failure in one of your factories that shut down production in that factory for a week, but your other factories were able to ramp up production so all your distributors and major customers noticed was a half-day shipping delay on some parts resulting in their own inventories, but your other end users didn't notice anything, then all you need to do is apologize for the inconvenience and say is that a plant had to be taken offline and it took half a day to add shifts to the other plants and get your widgets shipped out. If you are a public company you may need to issue a press release for the benefit of investors. If you had temporary layoffs or if employee health and safety were affected, you may have to notify the goverment, unions, and affected employees. Other than that, you probably don't need to say much more.