Why is ''à'' used in one place but ''de'' in the other?

A difficult area ! 

'Il/elle est' and 'c'est' are special cases in use of the prepositions à or de before the infinitive.

According to the attached information from Laura Lawless, with il/elle est or c'est, it varies depending on whether the pronoun is referring to a 'real subject' - that is a previously known noun that the sentence is remarking upon (effectively "specific/variable use") - or to a 'dummy subject' (effectively "impersonal/invariable use") for the verb.  I expect that like me, you will need to read the article more than once.

So, in your first example : 

the second sentence could be written, without using 'il', as "Ce verbe est très difficile à apprendre", as 'ce verbe' is the true subject of this sentence. In this sentence 'il' is a specific pronoun.

whereas in the second example : 

the second sentence cannot be replaced by "La comédie était difficile de rire", as 'la comédie' is not the subject of this sentence. In this example, the 'ce' is an impersonal pronoun (note: just for a bit of added difficulty, as impersonal expressions, 'c'était' and 'il était' are interchangeable here, but this is not the case for specific use when il or elle is needed as approriate). 

https://www.lawlessfrench.com/grammar/using-impersonal-expressions/