Politically, this is 'dynamite' for the case the FBI has been making for the last several months for creating a backdoor in encryption. Up to this point, the FBI has made the case using the media with the goal of having state or federal laws passed to create this backdoor. This case gives them the ability to 'speed up' the process by having this decision made in the courts, rather than having to slowly work through an increasingly stagnant legislative system. There is also plenty of evidence that this is an attempt by the FBI to make an example out of Apple. If the FBI can strong-arm what is currently the most profitable tech company in the world - a company that markets its products based on quality and security - into creating a backdoor into its products, then all the other tech companies will more likely 'roll-over' and put in back-doors as well. Also, we have evidence from several whistle-blowers and tech companies that the FBI is known to intentionally modify technical equipment to perform surveillance, so for them to go ahead and create the backdoor themselves for this particular iPhone is not exactly outside of their capabilities. In addition, Apple specifically requested that the FBI not go public with this case, but the FBI did so anyway, which is probably one of the reasons why the customer letter was written.
Technically, a backdoor in encryption is just a flat-out bad idea. Encryption is used for online shopping, online banking, healthcare records, and government secrets, just to name a few. In the case of the iPhone, this encryption is used to help ensure that if someone steals your phone, and you have it locked, then there is no way for them to access the data on the device. It is also used to ensure that messages sent between phones cannot be intercepted and modified between the sender and the receiver. The obvious downside is that it also can prevent law-enforcement from legitimately accessing the phone of a convicted criminal. However the idea of developing encryption that is strong for government but weak for individuals is flawed. In fact, there have been a number of recent computer attacks that happened because certain security technologies which were compliant with the now rescinded rules from the 80's and 90's against exporting encryption technology. The best analogy that I can think of putting a backdoor in encryption is the scene in the Lord of the Rings movie where Denethor says the Ring "should have been brought back to the citadel to be kept save, to be hidden dark and deep in the vaults, not to be used, unless at the uttermost end of need." The FBI would be creating something in order to be "safe", and yet would set up the very weakness that terrorist organizations and other bad actors would love to exploit, as there would be no way to stop them, since the hole in our defensive structure would be there by design.
Philosophically, the whole question is being placed by the FBI on the false dilemma of 'security versus privacy'. Apple is attempting to shift the conversation to more 'freedom versus fear'. The idea of 'security versus privacy' is a false dilemma because security is impossible without privacy and vice-versa. In order to establish the common good, respect for the person requires that his privacy be safeguarded, and that the security of the community is maintained, (see Catechism of the Catholic Church 1907-1908) To sacrifice one for the other would be destructive to the common good and would prevent either from being realized at all. Apple's attempt to re-frame the question is definitely closer to what we should be concerned about, but I do acknowledge that while they say the right words, the meaning behind them is very different, being rooted in a more individualistic sense of freedom, which I talk about a little bit here and here.
For a more in-depth look at this story, I highly recommend Troy Hunt's article: Everything you need to know about the Apple versus FBI case.