What healer skills were removed/changed with Shadowbringers?

[Saefinn]

I think a better representation is for 4.0 and 5.0 as stated the changes started in 4.0 and were doubled down in 5.0.

As I was a SCH, I'll give a bit more detail on theirs:
Eye for an Eye was a SCH only skill that became cross-class and is now completely gone. This was a defensive buff that could be spread with Deployment Tactics. The way this worked was that it'd have a 20% chance to lower the enemy's attacks by 10% when hit.
Shadowflare - this was shared with SMN this was a ground AoE DoT
Miasma II - this was replaced by Art of War, it was a DoT rather than burst damage
Miasma - This was a DoT
Bio -> Bio II - It used to be that Bio and Bio II were separate skills and would stack.
Selene's abilities, which include: Silent Dusk (A Silence skill), Fey Wind (an AoE haste) and an AoE Esuna called Fey Caress.
Once upon a time faeries also used to do a damage mitigation spell (back in 2.0 and 3.0)
Leeches was replaced by making Esuna a cross class ability. It made no sense, especially as they tied lore to Leeches in the SCH story and you had to use it and there was literally no benefit to removing it. It was just flavour.
Rouse -> This was a faerie buff. As somebody mentioned Seraph technically does this, so in effect, they repackaged an existing skill.
Both Miasma spells applied "disease", which would reduce the amount of healing an enemy could take and would slow their movement speed, it made Miasma II good for big pulls.
Virus - This was a cross class skill for healers but SCH got a version called "Supervirus". Virus would reduce a target's Strength and Dexterity by 15% and Supervirus would include Intelligence and Mind. This was repurposed as "Addle" and given to magic DPS but only affects Intelligence and Mind and it's a 10% reduction instead.
Bane - which like its SMN counterpart, it spreads your single target DoT's (Bio, Bio II, Miasma)

But wait, there's more.
Once upon a time some of SCH's abilities applied their own status effects, I've already described Miasma's, but Ruin II got a "Blind" effect and Shadowflare had a "Slow" effect.

You could also micromanage your pets differently. Your pet abilities were on your Pet hotbar instead of as abilities in your list. Whilst you could still execute them, the way they were executed meant there were completely independent of the caster, so for example if you were mid casting "Adloquium" you could also cast "Whispering Dawn". But you could also case "Embrace" this way without relying on the AI, so it meant weave kind of play to SCH, which was especially useful back in the days SCH's pure heals were weaker. But thanks to faerie AI, this was an optional playstyle. So as one who loved this approach it wasn't stepping on the toes of those who didn't.

DPS wise, you could think of SCH as a DoT mage, which I liked because I felt this was how it tied mechanically with its FFXI counter part (which worked differently, but still, it was a DoT focus). At its prime you'd have put down: Shadowflare, Bio, Bio II, Miasma and Miasma II, you'd Bane Bio, Bio II and Miasma. This has all been reduced to 1 skill, Biolysis.

Other things to consider is that SCH was a weaker "pure" healer and depended a little more its "shield" healing aspect & anything the mitigated damage and faerie use was more important and a choice. In those situations you needed to heal less, you'd use Selene who'd provide more utility, which is great because you take a healing loss.

So when you reflect back, SCH has lost a LOT of its flavour.