The first benefit of the War Caster feat is (PHB, p. 170):
You have advantage on Constitution saving throws that you make to maintain your concentration on a spell when you take damage.
The spell sleet storm doesn't deal damage, but the game designers still wanted it to make characters in the AOE make a Con save to keep concentration. So the spell description states:
If a creature is concentrating in the spell's area, the creature must make a successful Constitution saving throw against your spell save DC or lose concentration.
Does the War Caster feat still give you advantage on a Constitution saving throw (to maintain concentration) made because of the sleet storm spell?
I definitely don't think this is an intended interaction, but RAW, War Caster seems useless for this situation, since it says "when you take damage", and sleet storm is not a damage-dealing spell (the Con save to maintain concentration due to the spell isn't due to damage).
Best Answer
War Caster only helps with concentration saves caused by damage
The rules on concentration in the PHB list several ways that a caster might lose concentration, of which two rely on failing a Constitution saving throw. The first of these is as a result of taking damage, while the other is due to environmental phenomena (PHB, p. 204):
The first benefit of the War Caster feat states (PHB, p. 170; emphasis mine):
By including the text I've bolded, the feature makes clear that it only applies to concentration checks made from taking damage. If War Caster was intended to apply to all Con saves made to maintain concentration, it would have omitted the bolded text. As such, it does not apply to Con saves made to maintain concentration that are caused by environmental phenomena.
Sleet storm provides a specific rule for causing concentration checks outside of dealing damage, and so is also exempt. This fits with one of sleet storm's main functions as a spell that is good for breaking concentration.
You can compare this to the new Eldritch Mind invocation, whose description states (TCoE, p. 71):
This makes no mention of taking damage, and so would grant advantage on a Con save made to maintain concentration as a result of sleet storm.