It might be a matter of voice and presentation. Sentence 1 is speculative, and Sentence 2 is informative. It might not be grammatically incorrect, but it would be sub-optimal if the point of the sentence is to inform and not to give opinion.
Hiram Bingham first crossed the Andes Mountains in February 1909, the wettest month of the year, which should have made his travels difficult.
Here, a speculation is being made that the fact that February in 1909 was the wettest month of that year was the reason that his travels might have been difficult. However, this is not presented as a fact.
Hiram Bingham first crossed the Andes Mountains in February 1909, the wettest month of the year, making his travels difficult.
Here, a statement or conclusion is being made that February 1909 being the wettest month of the year was absolutely the reason that his travels were difficult.
Whether the information is true or not, I don't know, but this is the only flaw I see with this pair of sentences.