I apologise if this has been asked and answered (I'm sure it has) but after a decent amount of searching I can't find it.

Mage Slayer states "You have advantage on saving throws against spells cast by creatures within 5 feet of you."

Does that include concentration saves?

Say I'm a paladin, with the feat in question, getting up close and personal while maintaining one my host of concentration-dependant spell options and a caster manages to squeeze off a pot shot at me before I inevitably smite the everloving snot out of them. Taking damage from the spell forces me to make a concentration saving throw. Do I get advantage on said throw since I'm within 5 ft of the caster?

No a concentration save is not a save against a spell. That is what the warcaster feat is for.

But a spell forces you to make a saving throw. It's there a finer distinction I'm missing?

Quote from shadow1784 >>

But a spell forces you to make a saving throw. It's there a finer distinction I'm missing?

 But it is not a save against the effect of the spell. A concentration check is caused by taking damage not by what caused the damage.

Understood, but I would contend that the concentration save (not check) *is* an effect of the spell. I just think it's ambiguous and wanted to see if anyone else agreed or if I'm just off my rocker. 😄

As DxJxC said, “saving throw against spells ” are saving throws that a spell says you need to make in an attempt to reduce the effectiveness of the spell. Most concentration check con saves are not called for by spells explicitly, but a result of damage. 

But, there are a few spells such as Sleet Storm that call for a con save for concentration as part of the spell. For those, I would interpret that Mage Slayer helps, but I’m not too sure. Maybe some others might provide insight. 

Personally, I look at it as a spell cast against you is something that is trying to do something malicious towards you. A concentration save is, indeed, a function of the spell, but it is made for the spell, not against it.

I don't know if I'm being a bit pedantic here, but that's how it makes sense in my head.

In the example you give, you would get advantage against the saving throw of the spell cast by the creature that was within 5 feet of you - so a better chance to take half damage (thus lowering the potential DC of the concentration save) or resist its effects. The concentration check for (not against) your smite spell would fall under the purview of the Warcaster feat, as DxJxC mentioned.

Quote from Akacen >>

I don't know if I'm being a bit pedantic here, but that's how it makes sense in my head.

 Every rule discussion is, by necessity, pedantic. 😄

I understand what everyone is saying and can definitely get behind that interpretation. I tweeted Jeremy Crawford so maybe it'll get added to the annals of Sage Advice...

One other notable difference between concentration checks and spell saves (with the exception of spells like sleet storm) is that the concentration check isn't based on the save DC of the caster.  The mage slayer feat lets you observe an enemy closely and ready yourself for their spell.  If your preparations work, as evidenced by a successful save, any resulting concentration check will be easier as a result of the lower damage.

Concentration saves are generally only brought on by taking damage, or things that specifically trigger them like the aforementioned Sleet Storm spell.
Mage Slayer does not grant advantage on general Concentration saves, because those are triggered by taking damage and the DC is based on that damage, not by the Attack/Spell itself.

I would personally say you do have advantage on the Concentration save from the Sleet Storm spell if you are within 5 feet of the caster.  In the case of it, you're making a Concentration save against the spellcaster's Save DC as a direct result of the spell's effects.

A concentration saving throw is not a saving throw against a spell. They are categorically different things.

Mage Slayer only grants advantage to a saving throw made directly against a spell that targets an ability score. I.e., you have advantage on the saving throw against the DC set by someone casting Sacred Flame on you, but not against Guiding Bolt because that targets your AC.

A concentration saving throw (if you're actually concentrating on an ongoing spell effect at the time) is called for in response to having taken damage from any source. It does not matter whether the damage was from a spell or a mundane arrow. It is not a saving throw against a spell. Just as Mage Slayer does not grant advantage to death saving throws--even if the action which took you to 0HP was a spell--it does not  apply to concentration saving throws.