What is an example of a Constitution Check v/s a Constitution Save? Does anyone know?

Constitution check example would be holding your breath underwater. 

Constitution save example would the saving throw of a spell. 

Ability checks would apply to this spell, and saving throws would not.

It's also entirely possible that the DM will ask for a Constitution check with a skill proficiency that isn't normally linked to Constitution, due to the particular circumstances. For example, Constitution (Athletics) to run a marathon, Constitution (Stealth) to hold a concealed position for a long time, or even something like Constitution (History) for late-night research in a race against the clock.

... of course, many such examples would take longer than the duration of the spell.

The new "Oath of Heroism" paladin receives this as an always-prepared spell.

Combined with their bonus action Channel Divinity, which gives them advantage on Strength & Acrobatic checks, the Heroism paladin could use this to gain advantage on THREE different stats.

I'd wager Constitution, which makes the Heroism Paladin a physical powerhouse.

As DM I would have this spell grant advantage for any check with the given ability at it's core, including save and skill checks.

When upcasting this spell to target multiple targets, may different targets be given different effects? For example, could it be used to give both the party's fighter Bull's Strength and the Party's rogue Cat's Grace in the same casting?

From my initial reading of the spell I believe it is allowed but I wasn't 100% positive.

Does the bull strength ability stack on top of the oath of glory paladins channel divinity which also doubles the carrying capacity

I'm surprised this isn't a wizard spell

Quote from JohnEzell >>

What is an example of a Constitution Check v/s a Constitution Save? Does anyone know?

 An example of a Constitution check would be to sustain a forced-march pace, while Constitution saves are often made to resist a poison or disease.

Kinda sad there aren't any extra bonuses for enhancing the mental ability scores like there are for the physical ones :/

There are spells that require concentration and when it gets broken,  a CON save is used.   Otherwise,  it's alot when something is affecting by the body: temperature,  poison, stamina checks, etc.

Quote from JohnEzell >>

What is an example of a Constitution Check v/s a Constitution Save? Does anyone know?

There are spells that require, like this one, concentration and when it gets broken,  a CON save is used to try to maintain it.  Otherwise,  it's alot when something is affecting by the body: temperature,  poison, stamina checks, etc.

Quote from Stardust_Strifer >>

Kinda sad there aren't any extra bonuses for enhancing the mental ability scores like there are for the physical ones :/

 I initially thought that also, but then realized how many more skills there are for the mental ability scores.  No one is asking "What's a example of a Charisma check?" because we've all had that Bard in our party at one time or another.

This spell and the new Ability for 5th Level Sorcerers, Magical Guidance, from Tasha's. You can have three chances to succeed on any Ability Check.

with tasha's you can turn on the expanded spell list for wizard and get it. 

Quote from ravensburgermaster >>

I'm surprised this isn't a wizard spell

 It is now! they added it to the wizard spell list in Tasha's i don't know why it's not updated on here.

Enchance ability (int) gives advantage on counterspells and dispel magic if you are an INT caster (wizard). Ditto for charisma for bards etc.

Artificers get this at level 6.  During downtime, you can cast this 2 times per day (3 if you have a Spell-Refueling Ring infused,) so you can use this spell to learn a language or proficiency faster.  

XGtE says it takes 10 workweeks to learn to use a new tool or a new language (minus your Int modifier,) so use this spell to cut that time in half and be proficient (with artificer expertise) with ALL SETS OF TOOLS. 

Your push/lift/drag capacity is double your carrying capacity so would your push/lift/drag capacity increase as well if your carrying capacity is increased by the Bulls Strength effect?

If this doesn't boot skill checks this spell is pretty much useless