use of the articles

They are both correct.

Consider the reasons for using definite or indefinite articles. Use "one difficulty followed the other" when you are talking about already specified difficulties and "one difficulty followed another" when talking about difficulties which are so far unspecified.

Good God I wish I cold reason like Cen
But it's Alzie's don'cha know

No no Cen I mean me

(1) During clearing the back yard, [first] we found a dead dog, my mother-in-law arrived drunk, closely followed by the police, and [finally] I broke my ankle. One difficulty followed the other as the afternoon wore on.

First we list the difficulties, then we state that they happened one after the other. Because the reader already knows about them, we use the definite article, 'the' followed by 'other'.

(2) During clearing the back yard, one difficulty followed another as the afternoon wore on: [first] we found a dead dog, then my mother-in-law arrived drunk, closely followed by the police, and [finally] I broke my ankle.

This form is, perhaps, more usual. First we state that a number of difficulties were encountered, but because the reader does not yet know (might never know) what they are, we use the indefinite 'another'. Then we choose to list the difficulties, but we don't have to.

Thanks a lot, Centrox!