-지 말다 is primarily used with verbs since it means "do not (do something)". It can also be used with a subset of adjectives (or "descriptive verbs") that describe emotion, for which the -아/어 하지 말다 form must be used. As you may know, these emotion adjectives describe how a person feels, and takes a different form depending on the person of the subject. For the first person "I", a base form of the adjective is used (네가 좋다, 무섭다), while -아/어 하다 is used for the second and third person to describe reacting with such an emotion (너를 좋아한다, 무서워한다). Since -지 말다 is primarily used in imperative mood (there are some exceptions though) directed to the second person, we must use the -아/어 하다 form when using the emotional adjectives. Also, we cannot use -지 마 or -아/어 하지 마 for adjectives describing a general trait or state, because we cannot control them at will as the verb 말다 suggests. For example, neither 뚱뚱하지 마 nor 뚱뚱해하지 마 makes sense because 뚱뚱하다 (be heavyset) is not something you can choose not to be according to your wish. As for 늦어하지 마, it is because 늦다 is a verb that you don't see the -아/어 하지 말다 form. 늦다 can easily be mistaken for an adjective, but it is actually a verb for "to arrive or act late". You can see that it is a verb in phrases like 수업에 늦는 사람은 감점을 받는다. Adjectives can't take the -는 ending which signifies an action happening in the present or near future.

In the KR adjectives, when one describes the third person, i.e., someone other than the speaker, one needs to use the format "~아/어/여 하다". [N.B.: This can be very confusing, but language learning always contains a lot of rules and exceptions.] In consequence, when it comes to the prohibition form, i.e., "~지 말다", then naturally for description of the third person, one needs to use the format "~아/어/여 하지 말다". === 좋은 하루를 지내세요.／I wish you a good day.