I am not sure of the community's preferred approach here. I love the basic thrust of Michael D's answer, but dislike the attempted and unwarranted assumption of numerical quantification: English has ways to describe differences in degree that make no claims to numeration. In fact, numerical quantification in such circumstances is often purely metaphorical. Moreover "not as worse" is itself a grammatical error.
I do not want to downvote Michael's post because the basic thrust is spot on. I do not want to make massive edits to someone else's post. Thus, I have decided to post my own answer, admittedly modeled on his answer. I shall delete my answer if Michael amends his to address these concerns.
He is well again
means he has completely recovered.
He is less ill
means he has recovered only in part but is completely silent on degree.
He is better
means he has recovered but is completely silent on whether recovery is partial or complete.
In practice, if "He is better" is not further qualified, it may be inferred, perhaps incorrectly, that recovery is complete. But the bare statement does not formally imply that conclusion.
Also in practice, if "He is less ill" is not further qualified, it may be inferred, perhaps incorrectly, that recovery is far from complete. But the bare statement does not formally imply that conclusion.