Director James Mangold answered this in an interview with Empire Online.
In this XCU version, Logan's healing factor abilities have altered over time. While they still heal him, they no longer heal him perfectly, leaving scars.
“One of the things we all thought about as we worked on this film is, well, we don’t want to rebuild everything. We want to have some questions. In order to make a different Logan, and a different tone of a Wolverine movie, we felt like we couldn’t hold on to every tradition established in all the movies religiously, or we’d be trapped by the decisions made before us. So we questioned whether Logan’s healing factor causes him to heal without even a scar. We imagined that it may have when he was younger, but with age, he’s getting older and ailing. Perhaps his healing factor no longer produces baby-soft skin. So we imagined he heals quickly, still, but it leaves a scar. The simple idea was that his body would start to get a little more ravaged with a kind of tattooing of past battles, lacerations that remain of previous conflicts.”
In the final cut of the movie, his healing factor not only just leaves scars, it also works MUCH slower. It is heavily implied that the adamantium inside him is poisoning him, resulting in a decreased healing factor.