Or, "Why All Y'all Need to Stop Making This Comparison, PRONTO." (This is also the TL;DR)
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On the face of things, Lyse and Wuk Lamat seem to be similar. You, I, and everyone in between seems to have made this comparison within the last two weeks.
Lyse has suffered at the hands of the player base's ill-conceived vision of her for long enough. It won't end today, but by God, it should.
We'll start with their differences.
Wuk Lamat is a character who was introduced in an x.55 patch, specifically in 6.55. As the development team needed/wanted to start up new ideas, they took a recent fan favorite character who they likely didn't plan on being a big and popular long-term face for the story, that character being Erenville, and they stapled Wuk Lamat to him. With big bright crayon letters claiming, "This is Erenville's childhood bestie." They had Erenville ask the WoL to help Wuk Lamat. You are then treated to her personality, which turns out to be the same personality and mannerisms she will displayed all Dawntrail long. She's full of bravado, pretty good in a fight, and likes to eat. Erenville finds her insufferable, but feels some sort of endearment towards her. We had to wait and see how she'd play out, since there was nothing established about her until the moment of her introduction, but we are told and know from trailer material that she will be a central character in Dawntrail. We are implored to go, because her brothers are bad men who want the throne to start wars of conquest, though there was also the small fake sincerity around the, "It'll just be a fun adventure, c'mon, that's what you live for!" sort of ploy.
Lyse is a character who had been with us all along under an assumed identity, that identity being Yda. The facade being dropped in an x.55 patch, 3.55 to be precise. This was when Papalymo's, her chilldhood bestie/father figure, aether faded out of the world completely as Shinryu broken free from his sealing spell. It is then that she is revealed to actually be Lyse, who was the actual Yda's baby sister. As Yda, for most of 2.0~3.5 she'd been a ditzy hand to hand fighter who played the comedic klutz to Papalymo's straight man. She was close with Moenbryda. She was really good at fighting, able to fight Livia sas Junius to a standstill, and in 1.0 the only Scion able to lay a strike on Gaius van Baelsar. So, apparently, the Scions were in on it while the WoL was not, so for the first time we're seeing this character's face, and of course, this was done because they wanted to use her as the central deuteragonist for Stormblood and really flesh out her character and use it to tell the story/showcase the story's themes. There was no small contention around this, as Yda fans felt "betrayed" due to forming parasocial attachment with the mystery around Yda's masked face and ditzy persona. The worry became that she would be a totally different character, and she also lost that, "enigma allure" from no longer having her face covered. The change of name and backstory death of the real Yda, having happened a year prior to 1.0, has been stated to be hamfisted and pointless by hundreds of fans over the years. However, in hindsight, it is actually just a trope that CBU3 or whatever they're calling themselves now, likes to make use of, in that is the inheritance of identity as the carrying on of legacy. Some other examples of it in FFXIV include: Nael Deus Darnus, Shiva, and Azem. In FFXVI, Cid. Anyway, this leaves us with Lyse who is still a Scion, in spite of not really being an Archon or Yda, who has fought alongside us in her guise, asking us to go along with her and the other Scions to bring the Eorzean Alliance into amicable terms with the Ala Mhigan Resistance.
So right off the bat there's a lot more going on with Lyse than there is with Wuk Lamat. In some ways this hinders her, but in many others it justifies her incoming scenes and relation to the plot. The end of HW and beginning of Stormblood did this quite organically for Lyse, whereas the end of EW and beginning of Dawntrail do it quite artificially for Wuk Lamat. So let's move into their differences within their respective expansions now.
Spoilers for Dawntrail Ahead
For Dawntrail and Wuk Lamat, we take a sailing ship from Sharlayan to Tural. This ship is outfitted with a forcefield generator to protect it from rough waters and storms. It is also outfitted with the most incompetent crew of sailors known to man. Proper stowage of shipboard items is not done. No one ties themselves to the mast while working on deck during the storm. The WoL and Scions have to save the ship, and Wuk Lamat kind of contributes but mostly gets seasick. This experience is supposed to portray a rough and long sea voyage where we bond with Wuk Lamat, but it's under cut by the fact that in earlier expansions, long sea voyages are reduced to a shorter, indeterminate time thanks to most sailing ships having ceruleum engines on board.
Once we arrive in Tulliyolal we are shown around the city by Wuk Lamat, and told a little bit about its immediate surroundings. We stop at the Northeast gate so Wuk Lamat can tell us that the North America analogue is up that way across a vast bridge constructed by giants, and we must be dying to go adventure there, and promises to get us a pass to do just that once she's Dawnservant. We didn't ask or make that remark, she told us what we want to do even though we just arrived.
We go for an audience with Gulool Ja Ja who is sparring with Estinien to showcase he's strong. He informs us about the Rite of Succession, and we soon come to learn that Wuk Lamat is the scrappy underdog with no real platform for how she wants to rule, other than she just wants peace to be maintained, and as a result, almost none of the Tulliyolallans think her bid a serious, viable one. The other three contenders are her two brothers and Bakool Ja Ja. She has undefined pre-existing beef with Bakool Ja Ja, and he is shown to be a bad guy, because he stepped on a bag of tacos that she dropped.
The Rite will consist of 7 tasks, from each tribe of peoples that make up Tulliyolal and south Tural. The first task is to define and solve the problem the Hanu Hanu(Turali Vanu Vanu) are having with their crops. The crops are stagnant and dying, though they appear to have everything that plants crave. Racking their brains, the team comes to the conclusion that it's likely some sort of aetherial stagnation, since the wildlife in the fields is also sedated, more or less. Before anyone can suggest a method for fixing this, Wuk Lamat declares that we should help the Hanu Hanu with their broken festival float. This will liven up the Hanu Hanu's spirits, and perhaps get us more information. You're given a dialogue choice here, and one of the choices is to basically try and herd Wuk Lamat away from a seemingly irrelevant idea. Her reaction? She ignores you, proclaims her idea is a good one, and the Scions droll on that we have to let Wuk Lamat try her ideas because the trials are for her. Of course, her inane suggestion turns out to be a dramatically excellent suggestion, not because she knew anything about Hanu Hanu or Turali culture or anything, it just so happened that the festival float is actually a sorcerer's device for returning aether into the land to keep the crops fertile. We do live on Etheirys, after all, and Aether is king.
I almost don't need to go on describing or summarizing the rest of the plot of Dawntrail nor Wuk Lamat's role in it. This is essentially the whole expansion in a nutshell. Wuk Lamat faces adversity, and then she overcomes because the scriptwriter says so, not because she is a well written character who is overcoming herself. That challenge, too, is something the scriptwriter has solved by a single campfire pow-wow with the Scions telling Wuk Lamat she has to be more mature and honest about herself in order to rule and win the rite. Get in the backseat WoL, this is the Wuk Lamat show now! Your status as the WoL and savior of Etheirys are beyond the scope of this expansion, so speak when spoken to, and adventure where you're told too.
We run into the same thing again and again. Plucky Wuk Lamat is front and center and prevails. What was once insurmountable is effortlessly shaken off. When the story shifts away from her, the scriptwriter makes sure that people are still wondering about her and shifts it right back to her. There are other characters in Dawntrail, and they exist merely to pay tithes to Wuk Lamat.
*ahem*
As for Lyse and Stormblood. Stormblood begins after the Omega and Shinryu aerial battle. Lyse has revealed her real identity, and the Scions are called upon by the Eorzean Alliance to broker an alliance with the Ala Mhigan Resistance. In spite of us actively being a Scion ourselves, Lyse still asks if we are willing to come to Ala Mhigo with her, to fight for Ala Mhigo, or at least the trouble that gets in her way. Our dialogue choices are, "As if you had to ask!" or "..." I will be honest and say I don't actually know what happens if you pick "..." though it is most assuredly a, "But Thou Must!" situation. The active threat of the Empire looming over Eorzea, and all of the other Scions making this their mission in life. Would the WoL really refuse them all? Anyway, you go into Ala Mhigo lead by Lyse, who shows you to the Resistance's secret base, hidden by glamours on its entrance. She knows where it's at, because from 2.55 until about 3.4 she was working off screen for the Resistance along with Papalymo. You are then treated to your first encounter with The Ala Mhigan Resistance, which includes a couple of "familiar" faces, one being M'naago who'd arrived with the message of Baelsar's Wall and what not almost dead on the Rising Stone's door. The other being Meffrid, a forgettable NPC from the level 26ish MSQ from ARR dealing with Ala Mhigan Refugees in South Shroud. They vouch for your group and Lyse vouches too, and soon that amicable alliance starts to form up. You are sent with Lyse and Meffrid into The Peaks to Ala Gannha, in order to recruit fresh blood for The Resistance. Here, Lyse's Ala Mhigan heritage is called up for the first time in the expansion proper, when the village elder says she looks and sounds like her father, who lead an Ala Mhigan fight against the Garleans 20 years prior. Ala Gannha is a quarrying village, a stone gathering town, and it's been ground down under the heel of the Garleans for the past two decades. Lyse's ideals of fighting against the tyranny fo the fascist Garlean Empire are rebuked and reviled, because all that joining the Resistance seems to really get any Ala Mhigan is death. And she's further reviled for not being there in Ala Mhigo to suffer along side her people, as she was a refugee taken to Sharlayan by the actual Yda in her childhood. Lyse has no answer for this, and has to settle for turning away, showcasing that plucky idealism only goes so far.
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The actual difference between them is night and day. Lyse and Stormblood both explore far more nuanced circumstances, and she doesn't just brazenly have everything work out for her. People call her out. If she has a retort, she gives ones with her whole chest. If she doesn't, she is at a loss for words. She is a central character in the main narrative, where there are other stronger, better characters that somewhat overshadow her, and she is inspired by them.
Wuk Lamat and Dawntrail explore Wuk Lamat (and Tural a little bit). Every time the world building starts to get interesting Wuk Lamat is there to chime in about her Papa. Every time another character is reminded of how it fits into the larger world, Wuk Lamat is quick to dismiss that larger world and draw attention back in on herself.