schlimm can be seen as a weak augmentative of
schlecht, meaning it's a version of greater intensity. But the intensity doesn't need to be that much bigger. In many cases the english "bad" will still cover both. They are for the most part interchangeable. Sometimes
schlimm can take on the meaning of "awful" (which seems to be missing here).
Due to this small increase in intensity
schlimm can have, depending on context, a weak judgmental, but also empathic tone to it. (Not strong enough to be offensive though.) Whereas schlecht would remain neutral and objective or even mechanical.
A few examples to aid usage.
Du hast eine schlimme/schreckliche Handschrift.
You have bad/terrible handwriting.
Meant with slight judgement.
Du siehst schlimm/furchtbar aus.
You look awful/really awful.
Meant in an empathic way. E.g. towards a sick friend.
schlecht in those cases would feel as if you're uninvested.
The Duo example seems to fit well when talking about an injury or health issue.
Arzt: "Die Tests kamen zurück"
Patient "Ist es schlimm?"
Arzt: "Kann ich so schlecht sagen. Der Kollege hat eine wirklich schlimme Handschrift."
"Bad behaviour" translates to "schlechtes Benehmen".
"Benehmen " is an older word as "Verhalten". The difference is that "Benehmen" is more like your behaviour in terms of manners and "Verhalten" is like general behaviour. But don't misunderstand "Benehmen" as a translation for "manners". "The manners" = "die Manieren". The behaviour of a dog in respect of destroying pillows, barking all night or something like that is something that the dog should have been taught not to do, so it is called "Benehmen" here. On the other hand, if you are an ornithologist and observe the behaviour of sparrows, it is called "Verhalten".
Hope that helpfs to distinguish these two. ;)