It's an easy way to know what configuration options "this" aircraft has, long after it has left the factory.
Let's say that airline 1 ordered a bunch of 737's with one cockpit jumpseat & two aircraft batteries and the 3-hour cargo fire suppression option -- stuff you might expect for overwater flights. They got, hypothetically, the 737-7E1 (just making that up.... a 737-700 with customer code E1). Then airline 2 ordered a bunch of 737-700's with two cockpit jumpseats & a single aircraft battery and the 1-hour fire suppression system... a more short-haul sort of fleet. And let's say that their code was the 737-7E2. Years later, all of those aircraft from both airline 1 and airline 2, and perhaps others, have been sold to airline 3. Airline 3 has some 737-7E1's and some 737-7E2's. And if you're conversant with the codes in that fleet (as mechanics & others at Airline 3 would be), then you can see on a list that "this" aircraft is a -7E2 and so you know that it's the configuration with both jumpseats, one battery, and etc.
Those customer codes don't deal with things like the passenger cabin seating configuration; those can be swapped out pretty easily. But stuff like batteries & cargo fire bottles and plumbing for an extra O2 mask in the cockpit doesn't tend to change, so those sorts of things get captured in the customer codes.
As to why they abandoned them, my only guess is that they may have decided to standardize all their product by removing customizations -- a 787 is a 787 is a 787, and (perhaps) they don't let customers configure them in ways that, down the road, may make the aircraft less desirable for somebody looking to buy or lease a fleet that doesn't have lots of variants between jets.
There are stories from guys who flew freighters where ever aircraft on the ramp was different in some way -- even though they were all the same type. Newer INS, older INS, different flight directors, sometimes even different types of engines, etc etc. For a freight operation looking for any aircraft they can get cheap, that lack of standardization may be no big deal. But for other operators, not having to deal with the headaches of dissimilar aircraft in the fleet may be an incentive to choose "this" type over "that" one.