I looked up in the dictionary, 帮 means help, and 忙 means busy while 助 also means help, so is 帮忙 (help busy) and 帮助(help help) interchangeable? Or what is the subtle difference between them? Thanks!
帮忙 is more colloquial, it is intransitive or separable. 'A comes to help' translates into 'A来帮忙了' and 'A helps B' into 'A帮B的忙', not *'A帮忙B'.
帮助 is a transitive verb or a noun. 'A helps B' translates into 'A帮助B', the verb is not separated unlike 帮忙. 'With the help of A' translates into '在A的帮助下', usually not '在A的帮忙下'.
Another difference is that 帮忙 is always used for more concrete activities (related to its colloquial state). For instance, 'to help fight against infection' translates into '帮助对抗感染', never '帮忙对抗感染'. On the other hand, 'to help lift the box' translates into '帮忙抬箱子' more often.
帮忙->(verb+object) while 帮助 is a transitive verb or noun. I don't think it's interchangeable because their different grammar structure. 帮忙 is a fixed with an object"忙", for this we can call it 离合词. You cannot say 帮助[我的/他的]忙. However, semanticallay, 他帮我的忙。 has the same meanings as 他帮助我。(what kind of help not specified here).