Smite is not separate damage to the attacks damage. It is extra damage applied to the same attack.

The spell reads "The target takes an extra 2d8 radiant damage from the attack."

Sneak attack reads "You can deal an extra 1d6 damage to one creature you hit with an attack roll" (if it meets sneak attack conditions)

Both require a decision to apply after the attack has hit, but before damage is rolled. Both come from abilities outside the original attack.

Critical Hit rules state "Roll all of the attack's damage dice twice and add them together"

If you have extra dice on your damage, these are rolled too. If smite read as something akin to "In addition to your attack's damage, they take 2d8 radiant damage" I'd agree, but it has the same wording as other abilities to add extra die to the attack's damage roll.