催: to urge; to harry (someone to do something)
促使: to motivate; to cause; to prompt (someone to do something or something to happen)
'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 'to hurry', it is not the cause of '他快點出門'. the subject must have an intention to apply the verb 催
Note 2: there is no intention in '前蘇聯的威脅' to cause NATO to form.