As discussed in this thread, SGE is a much more mobile and smooth version of SCH. SCH has more power, but has to jump through more hoops to use it. For example, SGE can press Zoe then Eukrsia Prognosis then Holos or Panhamia (or both...), but basically 3-5 button presses, less than two GCDs in duration, making it extremely responsive. Oh, and the collective one GCD in there, Eukrasia Prognosis, is entirely instant casts so mobile. If you lead into this with Eukrasia Dosis or Toxicon or Plegma, you have 2 GCDs before Eukrasia (Zoe and either of Holos or Panhamia), and one after Prognosis (the other of Holos or Panhamia if you want it). Oh, and did I mention you can also layer a Kerachole on the front of that Plegma if you want? Because you can.
Contrast with SCH. While you CAN make a far stronger barrier, this is going to require some setup. You're going to want Recitation before Adlo, and Adlo is a hardcast. You CAN blow Swiftcast on it, but that's an additional oGCD weave on the front end. You need to be smart about who you use the Adlo on, though, so the shield isn't consumed before you spread it. You then have to weave Deployment Tactics on the back end. This also will clip, and there's no avoiding this unless you weave it after the next spell cast, and it's Biolysis or Ruin 2 OR you Swiftcasted the Adlo, in which case you had to double weave before or use the Recitation before even that GCD. You also can use Protraction on the target to boost the size of the shield. You will also possibly want to use Dissipation before this, so you need to make sure you aren't overcapping your AF, so before that you'll have needed to use excess AF. Maybe one weave from Soil, but depending on how recent your last Aetherflow refresh is, you may have up to 3 charges you need to burn first. Alternatively, you can forego Dissipation if you want to use Seraph, but you'll need to weave Seraph a few GCDs early so she can get over her summoning sickness to cast Consolation. But if you're using Fey/Seraphic Illumination, you'll need to make sure to do that BEFORE all that so it doesn't get ghosted as Eos departs and Seraph comes in, or you'll need to do it after she comes in, in which case you have to wait for her to come in, have summoning sickness, then cast Illumination BEFORE you cast the Adlo. And with all that done, NOW you can hit Deployment Tactics on your target that hopefully still has most/all of the shield and is centrally positioned in the party (though this is WAY less of an issue than it used to be), and congratulations, you now have the strongest defensive protection for your party in the game. Oh, and don't forget you also need to make sure you're hitting Chain Strategem on CD and you have some extra random abilities like Excogitation and Aetherpact and Whispering Dawn and Fey Blessing you can throw around if you need to. Oh, and you DID remember to preplace Eos/Seraph in a place where the party's benefitting from all her abilities for this mechanic, right? And if you need yet more protection - Expedience DOES have that damage mitigation in addition to that movement speed...
That is, of course, the over the top version, and you can do far less, but the point is that SCH has a lot more going on in how it's thinking and approaching these problems. SGE is a Barrier Healer and so has some proactive stuff going on with it, but it's far less and the abilities don't have "waiting times" or "lag times", making it much more sharp and responsive, though it pays for this by being weaker...but it's still completely sufficient for the task (no encounter will be designed to require an Infinity Gauntlet powered Spreadlo, otherwise it wouldn't be clearable for parties without a SCH, and that's not happening). Moreover, SGE's direct healing spells are more accessible and often more potent than SCH's. Zoe + Pneuma is Cure 3 levels of healing, and Physis is an oGCD Medica 2 that also boosts healing further. A single button press that does Fey Blessing + Whispering Dawn + (part of) Fey Illumination, and does so without ghosting or Eos lag time issues. And unlike SCH, SGE is incentivized to burn AG, meaning Kerochole is something you can use frequently for even more mitigation and healing while SCH is having to decide if it wants to sacrifice (a pitiful amount of) potency in order to access a ground placed version of the same effect.
...and these are the two Healer Jobs that play THE MOST alike.
WHM is completely different on one side of that, and AST is completely different on the other. Essential Dignity is actually used (unlike Lustrate and Durochole) as a spot heal, for example, and AST has a "healing plan backbone/scaffolding" in the form of Earthly Star. Some of its abilities have lag times that you want to factor into, such as Star and Macrocosmos. Essential Dignity is the spot oGCD with the most room for optimization (using it as close to 30% or less health on the target as possible, obviously without waiting for them to die, to increase its value). It MIGHT (or might NOT) have Lady up for additional healing (on par with Assize), but that's also yet another thing to keep pressing on CD (Minor Arcana draw) and react to/track. It has some powerful but niche spells that many people don't even use (Synastry, for example), can put on a Barrier Healer cap for about half a minute with Neutral Sect (once per 2 mins) and Collective (once per minute). And all this is going on with it maintaining Draw and Play with a cadence of 3 per 2 min and 1 at the 1 min mark in between, possibly trying to optimize with Redraw, keeping Divination on CD every 2 mins, and using Astrodyne at 3 played cards and not overcapping seals.
On the other hand, WHM is pretty much using Assize on CD, using Rapture (boosted by Plenary once per minute if a bit of extra healing is needed) for most healing, Asylum if it needs more, Tetra/Benison if it needs spot healing, and tapping Temperance for big damage needs or Lilybell for the "several waves of damage" stack marker. And if all that SOMEhow fails, it just falls back on Medica 2 and then Medica 1 or Cure 3 spam. It would have difficult to make WHM more different than AST if you tried.
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The Healers are only really similar in their damage kits. Their healing kits really are quite distinct and in some cases, all over the place, depending on what you're trying to do with them. Their damage kits are very similar (though not identical, with SGE and SCH both having different optimizations based on the GCD/oGCD + resource respectively), but while you can find parallels with various abilities (Eukrasia Prognosis is KIND of like Succor...if Succor was two buttons that were a mini 1 sec and 1.5 sec GCD and also instant cast...), most still work differently, and some are combinations of various other things.