>催: to urge; to harry (someone)

>促使: to urge; to cause; to prompt 

催 has to be with intention. e.g.  in "我催他快點出門" ( I urge/ hurry him to go out sooner). 'I' have the intention of 'urging him to go out sooner'

促使 has to be with a cause. e.g. in "前蘇聯的威脅促使北約成立" (The threat of the former Soviet Union prompted the establishment of NATO) '前蘇聯的威脅' is the cause of 'the establishment of NATO'

Note 1: '我' is the subject that applies the verb, it is not the cause of '他快點出門'. *You must have a subject when you use the verb 催*

Note 2: there is no intention in '前蘇聯的威脅' to cause NATO to form. and '前蘇聯的威脅' is not actually a subject, it is the topic, and '促使北約成立' is the opinion/comment in a [topic + comment] sentence structure. On the other hand, "我催他快點出門" is a common [SVO] sentence structure