Maybe. Another possibility for a "Rosetta Stone", or perhaps the Lovecraftian perversion of said stone, comes from the ADE situation [0][1]. Even category theory, that great unifier with its own "Rosetta" tables [2][3], cannot escape; quivers have an ADE property.
Either way, we still don't know what's up with 1728. That will truly unlock everything, I think. Understanding 2, 3, or 5 would be nice; understanding 8 or 12 or 24 would be groundbreaking; but I think understanding 1728 will also be understanding Langlands' programme entire.
They are different enough that it's frankly a bit difficult to make sense of that question. I think in general, when asking "what is the difference between X and Y" questions, it helps if you elaborate a bit on why you think X and Y are similar / difficult to distinguish. That gives the person who answers the question a bit of a starting point: they can talk how the similarities you perceive might not be the whole story, or might be illusory (and you might end up answering your own question while clarifying your thoughts on the matter). Otherwise it might not be clear how to even begin answering the question.
Reflecting on this more, my best guess is that you think they are similar because they are both a "grand unified theory of mathematics". To explain the difference extremely briefly then, I'd say it's a massive exaggeration to call either category theory or the Langlands program a "grand unified theory of mathematics." They each demonstrate connections between some areas of mathematics, but they are different connections, and fall very short of unifying everything. If you want to know more, I think you'd be better off learning more about category theory and the Langlands program independently of each other rather than looking for a comparison between them.
The two are not related in terms of people working in the fields. I have never heard anyone from my category theory circles mention the Langlands program and saw it for the first time on Wikipedia.
However, skimming the Wikipedia page, I can see a few interesting things, e.g. Galois groups and a notion of functoriality [1] that are analogous to concepts in category theory. It doesn't look like the Langlands program has been "Categorified" a la Grothendieck yet.
Either way, we still don't know what's up with 1728. That will truly unlock everything, I think. Understanding 2, 3, or 5 would be nice; understanding 8 or 12 or 24 would be groundbreaking; but I think understanding 1728 will also be understanding Langlands' programme entire.
[0] https://en.wikipedia.org/wiki/ADE_classification
[1] http://www-groups.mcs.st-andrews.ac.uk/~pjc/talks/boundaries...
[2] http://math.ucr.edu/home/baez/rosetta.pdf
[3] https://ncatlab.org/nlab/show/computational+trinitarianism