Since long time ago I have been thinking in two problems that I have not been able to solve. It seems that one of them was recently solved. I have been thinking a lot about the motivation and its consequences. Mostly because people used to motivate one of them with some very interesting implications. My conclusion however, is that there is a mistake in the motivation of the problem, and that, while being a really interesting result, it does not make any sense in the setting in which is formulated. As my opinion is not relevant compared to one of the experts in the area, I do not say anything. 
My question is if you can provide me some examples of conjectures that were believed to be interesting in the mathematical community because of a specific reason, but that once having the proof, people realized that the reason to motivate the problem  was not truly related to its solution. Or in other words, the solution of the problem gives no clues about the original motivation. 
I hope it was clear enough to understand the question. I could not find any such examples in the literature, so I assume that they are not that frequent.

Any hint would be appreciated!

Best regards,

Diego Flores-Hernandez

PS: I am assuming that most of you are not familiar with the two problems I am thinking about. So, I do not write them here. I hope this will not be a serious drawback
To give some context, here are my two examples:
1. The Tijdeman-Zagier theorem which says that $n^3 + n^2 + n + 1$ is prime for a positive integer $n$ if and only if $n = p^2$ for some prime $p$. 
2. The Fermat-Wiles theorem which proved that $x^{p} + y^{p} = z^{p}$ has no