I was reading both of them, and if you had a hypothetical Barbarian Paladin multiclass, would the use of divine smite be prevented by raging? Unlike other smites, it says that it expends a spell slot, but isn't listed necessarily as a spell. It reads more like a special ability that just happens to cost a spell slot.
Starting at 2nd level, when you hit a creature with a melee weapon attack, you can expend one spell slot to deal radiant damage to the target, in addition to the weapon’s damage. The extra damage is 2d8 for a 1st-level spell slot, plus 1d8 for each spell level higher than 1st, to a maximum of 5d8. The damage increases by 1d8 if the target is an undead or a fiend.
Rage: (Relevant portion)
...If you are able to cast spells, you can’t cast them or concentrate on them while raging.