You have asked a question about one of the earliest detective stories, upon which Sir Arthur Conan Doyle based his famous detective Sherlock Holmes. One of the key ingredients in this genre is the bumbling, well-meaning policeman, who is dismissive of the skills of private detectives such as Dupin in this story, and is determined that good old-fashioned police-work will figure out the crime. You would do well to think about how Poe builds this picture up in this short story.
It is clear that Poe is none to complimentary regarding Monsieur G. of the French Police. From the start there are somewhat dismissive remarks made about his limitations. For example, consider how the narrator comments upon Monsieur G's comment of Dupin's that he finds it easier to reflect in darkness:
"That is another of your odd notions," said the Prefect, who had a fashion of calling every thing "odd" that was beyond his comprehension, and thus lived amid an absolute legion of "oddities."
The inference is clear - there are lots of things that are "beyond his comprehension", and because he lives amidst a "legion of oddities" his ability to comprehend is obviously not that impressive or powerful.
However, it is clear that Monsieur G. does have his strengths. As Dupin begins to reveal how he gained the letter to his friend, he does say:
"The Parisian police... are exceedingly able in their way. They are persevering, ingenious, cunning and thoroughly versed in the knowledge which their duties seem chiefly to demand. Thus when G. detailed to us his method of searching the premises at the Hotel D., I felt entire confidence in his having made a satisfactory investigation - so far as his labours extended."
Dupin continues to support this by saying all the extensive measures that Monsieur G. took were "carried out to perfection" and that if the letter had been placed where they were looking they would have found it. However, Monsieur G.'s greatest failing in this case was his inability to think outside the box and consider other alternatives. This is what led to his failure to recover the letter - his lack of perception and comprehension.
So, whilst Monsieur G. is a diligent, hard-working Prefect who is good at his job, he lacks a crucial element of imagination and wider perspective that separates him from the greatness of Dupin. It is this fact that is recognised and exploited by the politician who has purloined the letter, and, in turn, it is Dupin's ability to do precisely what Monsieur G. is unable to do that allows him to solve this curious case.