I will post this in sections explaining the pro's and con's of the 2 classes and finally compare the 2 healer classes side by side. I don't have the time to finish this all at once so I will be posting it in sections/revising as time permits. Any good information will be appreciated/incorporated.
Overview for new players:
TICKS
First thing to understand is what a tick consists of. A tick in FFXIV is 3 seconds. Things to keep in mind are that most ticks meld together and synchronize, as in if you cast Aero I and Aero II on a mob, you will only see 1 damage number, but it is the total of the 2 dots. This does not, however, occur with things like the basic mana regeneration and shroud of saints, which remain separate. A standard mana tick is 2% of your maximum mana pool.
STATS
Mind is your primary stat. Gaining more mind is always more effective than gaining a secondary stat such as determination, spell speed, or crit. There is of course the exceptions like trading 20 determination for 1 mind or some such extremes. If people have a simple weight chart for healers I'd like to add it.
Determination affects both healing and damage spells. Though the gains are not considered to be huge, it is consistent.
Crit add's a small % chance to cause a +50% increase on a cast. It is especially important for SCH's and will be explained more in their section.
Spell Speed affects how fast you can cast a spell. It makes it easier to make swift recoveries, but it does not increase mana efficiency like CRIT or DET.
All healing spells have about a 3%+/- deviation, which means you don't always heal for the same amount. As you get higher level/more gear, the gap looks to be larger but it is simply because you're throwing out larger numbers.
WHM Overview
If the potency of something resembles say Medica II below, that is 200 potency + 50 Potency x10 ticks
HEALS
Spell Potency Cost Range
CURE I 400 @ 133 30y
CURE II 650 @ 266 30y
CURE III 550 @ 505 30y x 6y AoE
MEDICA I 300 @ 372 0y x 15y AoE
MEDICA II 200+50(10t)=700 @ 452 0y x 20y AoE
SS 18%(Target MaxHP) @ 266 30y
REGEN 150(7t)=1050 @ 186 30y
BENDICTION 100% @ 5min 30y
OFFENSIVE
STONE I 140 (20 second slow) @ 133 25y
STONE II 170 @ 159 25y
AERO I 50+25(6t)=200 @ 106 25y
AERO II 50+40(4t)=210 @ 139 25y
HOLY 200 (4 second stun) @ 532 0y x 8y AoE
FLUID AURA 150 (KB+Root 6s) @ 30sec 5y
REPOSE SLEEP 30sec @ 212 30y
BUFFS
PRESENCE OF MIND 2x SPELL SPEED for 10s @5 MINUTES
DIVINE SEAL 30% INCREASED Healing @1 MINUTE
SHROUD 212MP(5t) @2 MINUTES
PROTECT 15%ARMOR/MAGIC
PROS
Largest AoE heals in the game
Highest AoE damage(via Holy) in the game
Largest Single target pure heal
Easy dps rotation (ensure dots are on, spam stone II) is usually ~200dps with relic +1
Simple and easy to play. In early play it comes down to wack-a-mole. In later content it comes down to timing heals to land exactly after an enemy attacks.
Get Esuna early on for removing status effects
Enhanced Protect
Divine Seal
CONS
Mana Management A single ability on a 2 minute recast that restores a flat amount regardless of gear. Shroud restores 1060mp over 15 seconds. To add salt to this wound, you can also tack on that at lvl 48 WHM's get a trait that enhances Shroud of Saints. It does nothing.
Due to 2 of the class traits being 15% chances attached to Cure I and Cure II, it makes our management inconsistent. Some times you get so many free cure procs you never run out of mana. Other days you might get 2 procs over a 10 minute battle.
Aggro Issues When dealing with an under-geared/less experienced tank, WHM's tend to have aggro issues. With static groups or decently geared tanks this is a non issue, but healing a fresh 50 can end up being frustrating. The only deaggro we have is also tied into our only source of mana regen described above.
Damage Mitigation Stone Skin is based on your targets max hps. This means that as you gain more gear, the skill doesn't get better. If your target is seriously over geared then yes it helps, but if you have say a target that died once or twice, and you're attempting to protect them as much as possible they actually receive minimal gains from the skill, which means they can easily be 1 shot even with SS being active. While it is better than nothing, it still has minimal scaling outside of using it strictly on tanks.
Raid Utility Has 0 innate utility. Through the use of cross class abilities this is improved, but you get the weaker version of skills when they are cross classed.
Emergency Abilities The 2 emergency style abilities that WHM's get are Benediction and Presence of Mind. Both of these spells are on 5 minute recasts, which means that for most battles once they are used, they're gone until your succeed or you wipe, and even then they might still be down. There have been many discussions on Benediction and how most of the time when it is required or even planned for it still fails because of its 1.5 second animation cast. There have been many cases of target has 1400 hps, is healed for 6200 hps, takes 1500 damage, and still dies. Regardless if benediction succeeds or fails, it is always consumed upon starting the cast. Presence of Mind is only truly great if you build a spell speed gear set, which means that you decided to sacrifice the more mana efficient stats DET/CRIT.
I will be comparing the classes base on similar in purpose abilities. Some of them will come off as a stretch, but with some imagination and understanding of the game you should pick up on what I'm getting at.
COMPARISON WHM vs SCH
HEALING
Cure I vs Physick - These 2 ability are literally copy paste of each other. Cure I has an extra passive that gives a 15% chance for a free Cure II, but to give an example of how random it is during turn 4 today phases 1-4 I had 1 pull where I got 4 free Cure II's. That's 4 minutes of steady streaming of cure I. I also had 1 pull with 17 Free cure II's. Since the passive is completely random, it's not something you can rely on. It is sadly integral to WHM mana management.
Cure II vs Aldoquium - 650 potency heal vs 300 heal + 300 shield. Cure II has a small chance to proc a half cost cure III, which usual comes when you really don't need it. Effective use of Cure II to work with mana management relies heavily on Cure I procs(see above). Is great for recoveries especially if it crits or is used in conjunction with healing buffs(divine seal, convalescence, mantra, etc). Aldo can be used preemptively for shielding known incoming damage (death sentence, mountain buster, wicked wheels). It can also be used when a tank is knocked down heavily to buy time. Passive included on aldo is that crits double the effectiveness of the shield. IE if you were to heal for 100 shield would be 100. If you crit the heal would be 150 the shield is 300.
Cure III - This is a heavily costly heal, and is almost impossible to use outside of static grouping. Its group healing potential is outstanding and the scholar has no skill that matches it.
Regen vs Embrace - 150 potency heal every 3 seconds on the target that is was casted on vs 300 potency heal every 3 seconds that can be targeted on the fly or macroed to double up with other heals like physick or aldo. Regen costs mana, GCD, and to get consistent results has to be recasted on a target every 21 seconds. Eos can literally just decide to cast embrace for you, and effectively crushes any reason to ever want regen over embrace, yet I still see SCH's complaining about not having a regen ability. Regens can be double stacked and this has started some bad discussions about the double WHM combo vs double SCH. To that I would says if you both go Eos you are still technically double stacking, and doing it with a better ability.
Medica vs Succor - 300 potency heal vs 150 heal + 150 shield. Very much comparable to the description of Cure II vs Aldo except neither have a passive. Succor has the ability to be used preemptively, but lacks the recovery strength of Medica.
Granite Skin vs Stone Skin - 18% max life shield vs 10% - Neither of these skills are very mana efficient, and they both see their most use before a battle occurs. It allows players to take 1 free aoe for the most part. Helps with initial pulls, and early mana management. After that the WHM has to rely on granite skin to mitigate huge incoming damage abilities, whereas most SCH forget they have SS and just use Aldo.
Medica II vs Whispering(roused) *(iLvL 83)
DS + Medica II ticks for AVG 161 10x over 30 seconds = 1610 + 624 initial (2234 total) ~74hps / Costs 452 mana with a ~2.85 GCD/cast time can cause aggro issues(WHM death) depending on fight/tank
Rouse + whispering dawn does AVG 338 8x over 24 seconds = 2710 total ~112hps / Costs 0 mana, no GCD, fairy 1.9 second cast/may cause Eos a death depending on fight
Rouse + Fey Illumination + whispering dawn AVG 405 8x over 24 seconds = 3244 total ~135hps / Costs 0 mana, no GCD, Fairy has slight delay + 1.9 second cast//may cause Eos a death depending on fight
Simply put, the rouse + whispering dawn combo is better, because no fights currently require a consistent weak HoT. Medica II does have extra range, and is great for pug groups that have people that constantly stand at max range from everything. Medica II can also be double stacked like regen with the WHM/WHM combo. SCH/SCH combos that work together using communication and/or parser timers can however rotate the rouse + WD and can almost keep it up indefinitely at no cost to themselves.
Benediction VS Lustrate - 100% every 5 minutes vs "potentially" 25% 6x in the first minute and 3x every minute there after. These are emergency "instant cast" abilities that have 1.5 second cast animations. That means that they can both fail to save the tank, yet still be expended. If benediction was used early and then used again after 5 minutes it would heal 200%. If lustrate was used early, it could be potentially be used 21x = 425% in that same amount of time. Lustrate is a consistent emergency ability, and it is for most purposes much better than benediction. The exception is if the tank is about to take a massive hit next, lustrate might not be enough to save the day in those instances.
BUFFS AND UTILITY
Enhanced Protect vs Protect - WHM's give 15% magic resists over SCH.
(Eos)Fey Covenant - Gives party 20% magic resists castable every 2 minutes. Situational as almost all damage is physical, but the WHM doesn't genuinely have a matching skill. People will point to the skill above, but if you had a WHM buff your party with protect and then went into an instance with SCH/SCH anyways you could technically get the benefits of both worlds, while not using a WHM.
Divine Seal vs (Eos)Fey Illumination - 30% increased healing self buff 15 seconds every minute vs 20% group wide healing SPELL buff for 20 seconds every 2 minutes. Divine seal gives WHM's the ability to do amazing burst healing without actually using an emergency. It also helps greatly with mana management. Fey Illum assists both healers. as well as anyone else throwing an assisting heal, including the fairies. Running SCH/SCH and using communication can lead to using this to effectively give both SCH's the same effect as divine seal (once every minute).
(Selene)Fey Glow (spell speed) and Fey Light (skill speed) cannot be stacked together, but increase their respective stat by 30%. They last for 30 seconds and recast is 1 minute (selene normally just rotates back and forth between the 2). WHM's have nothing comparable.
(Selene)Silent dawn single target silence. instant cast, lasts 1 second, 40 second recast.
Virus vs Enhanced Virus - SCH's get -15% INT/MND over WHM. The benefits of the extra debuff is that you can bring down the damage of magic abilities.
Eye for an Eye vs Enhanced E4E - E4E is a great ability, that while not entirely consistent, is still incredibly helpful. The SCH can cast it every 2 minutes while WHM is every 3 minutes
(Erects a magicked barrier around a single party member or pet. Duration 30 seconds. 20% chance that when barrier is struck the striker will deal 10% less damage for 20 seconds.
Sacred Soil - A SCH only ability that creates a bubble that reduces the damage taken by group members to 90%. Has a 20% chance that the next succor will cost no MP. WHM has no comparable skill.
Shroud of Saints VS Aetherflow. 212MP x5(1060) over 15 seconds castable every 2 minutes vs 20%(At iLvL 83 thats ~880, at ilvl 90 its over 1000) every minute. SCH gets 1.5-2x the amount of mana a WHM receives. Aetherflow is instant, whereas SoS takes time to recover. Shroud contains a deaggro that is usually not required unless with an undergeared tank or at P6 of turn 4. Aetherflow grants 3 charges to the SCH to use their most powerful abilities. Those abilities, while limited, are free of MP cost/GCD.
Energy Drain - Requires the use of Aetherflow charges, can help recover mana when you don't have a bard. Usually not better to pick over lustrate, though if aetherflow comes back up and you have a remaining charge you can always just drain, then aetherflow and recover a massive amount of mana and fully prepared for the next phase of a battle. WHM has nothing comparable.
DAMAGE