They don't stop decomposing, but they do so rather slowly. In The Walking Dead, unlike the book World War Z, zombies steadily decompose, albeit rather slowly. In the book World War Z, the process still takes place, but usually at a far slower rate (the exception being the rare cases in which the zombie is in an extremely humid, hot environment, like a jungle; in this scenario, the zombie will decompose very quickly).
Getting back to TWD, the fact that the zombies decompose is particularly noticeable in the cases of zombies who have spent some time in the water. Their bodies become bloated and water logged, and their tissue becomes quite fragile:
The production team has also explicitly stated that as the seasons progress, they have tried to make the zombies appear more and more decomposed and emaciated. I have to admit that, in my opinion, they have been extremely inconsistent in this regard, but they claim to have taken it into consideration, and tried to reflect the fact that time is passing and the zombies are becoming increasingly decrepit.
It was most apparent in season 2, in which you really can see how skinny most of the zombies are. The casting director specifically selected people who were extraordinarily thin, and if I recall correctly, they had a lot of success in hiring marathon runners for exactly this reason.
Robert Kirkman has also said that as time goes by, the zombies lose much of their mobility, speed, and strength, as well as their already-meager amount of intelligence. This is actually how he explained the fact that in season one, and only season one (in fact, only the first two episodes of that season), zombies do things that seem to be beyond their capabilities - they almost run in some scenes, they use rocks to break windows, Morgan's zombie wife tries to turn a doorknob, and one little girl zombie stops to pick up her teddy bear. According to Kirkman, all of this is explained by the fact that the zombies are much more intact and undecayed, relative to the zombies of later seasons.
From a Reddit Ask Me Anything with Robert Kirkman:
Q: "...In the beginning of the show we saw walkers do things like using a rock to help bash the doors in or turning a door knob, is there a reason we've stopped seeing them do that?"
A: "Older zombies are less together and capable or doing things like that. Fresher zombies, which there were more of in season one, are able to do more than older, more rotted zombies."
If we're talking about TWD zombie bones as compared to real human bones, it seems clear that, although no one really talks about it, the zombies' bones, especially their skulls, are much less robust than our own.
As I said, no one seems to mention it on the show, but we can see it quite clearly. People routinely crush zombie skulls by stomping on them, hitting them with fairly flimsy objects, etc, and the most popular method of killing zombies is now the classic stab-in-the-head.
While I don't recommend that you try this at home, I can assure you that stabbing someone in the face won't usually result in the knife piercing their brain cavity. The skull is surprisingly thick, robust, and extremely tough. It is very difficult to penetrate it, especially with something like a knife or screwdriver.
The shape of the human skull - almost spherical in the upper portion - is such that a thrust with a knife is almost certain to be deflected and slide off to one side, which might inflict quite a bit of damage to the scalp and/or face, but the skull itself will suffer little or no damage whatsoever. Even if you managed to strike the skull perpendicular to the forehead, for instance, the knife might very well stop short, creating a possibility that you will injure your own wrist, or even worse, that your hand will slip forward and be sliced open by the blade.
As for stomping on a head, this might cause a concussion in a human victim, but it probably wouldn't crush the skull outright. And yet we see, again and again, zombie skulls popping open like grapes when the slightest amount of pressure is applied.
Moving on to blunt objects, such as baseball bats, rocks, clubs, and so on, again, the injuries inflicted upon zombies are far more serious than would be the case if the target was a living person. Yes, of course a powerful blow from a bat would fracture a human skull, but fractures aren't enough to kill a zombie. You need to destroy the brain, not merely create a few hairline fractures in the bone around it. Once again, attacks which would only hurt a person are consistently shown to kill zombies.
In season 3, we saw a zombie in the prison, whose hands were cuffed behind his back, snap his own arm in half with very little effort. This too would be almost impossible for a living human being to do. And in the season 5 finale, Rick was able to slowly shove a gun barrel through the roof of a zombie's mouth and into the cranial cavity. Also something that you couldn't do to a human without a tremendous amount of force.
We've also seen two zombies being killed by holding a relatively thin branch perpendicular to their heads, at about mouth-level, and pushing with very little force. Branches less than an inch and a half in diameter are not as strong as a human skull. The branch should break before the skull does.
And in season 4, when zombies began to fall through the ceiling of a big box store, a drop of about 15-20 feet, they literally popped when they hit the ground, almost like water balloons. A person who fell the same distance would probably break some bones, but they certainly wouldn't explode.
All in all, we are never told that zombie skeletons are far less robust than human skeletons, but all the evidence points in exactly this direction. Zombie bones are much more brittle and prone to breaking than human bones.
In a recent interview, Fear the Walking Dead showrunner Dave Erickson said that on that show, zombies will be more difficult to kill than they are on the original The Walking Dead series, because Fear the Walking Dead takes place at the beginning of the outbreak, whereas TWD is now at least a couple of years into it. As a result, zombie skulls on FTWD are stronger and harder to penetrate than has been the case on TWD.
Erickson said one of Kirkman's notes for Fear was that the zombies be different than how they were first seen on the original show. With the early onset of the outbreak, skulls will still be hard and the point, Erickson said, is to see a character trying to stab a walker in the head and not be able to penetrate the skull. "The point was it's hard physically to kill somebody," he said.
In fact, we've already seen this happen, on the second episode of the new series.