The paladin's Improved Divine Smite 11th-level feature gives every melee weapon attack an extra 1d8 radiant damage.
Their 2nd-level Divine Smite feature does 1d8 additional damage to fiends and undead when they expend a spell slot to smite.
Does Improved Divine Smite deal additional damage to fiends and undead as Divine Smite does, without expending a spell slot to use Divine Smite?
Improved Divine Smite says:
By 11th level, you are so suffused with righteous might that all your melee weapon strikes carry divine power with them. Whenever you hit a creature with a melee weapon, the creature takes an extra 1d8 radiant damage. If you also use your Divine Smite with an attack, you add this damage to the extra damage of your Divine Smite.
There is no mention of extra damage for fiends and undead so no extra damage is done (beyond the 1d8 extra it does to every creature of course). Contrast this with Divine Smite which says:
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.
Without that line of text there is no reason to think that Improved Divine Smite does the same thing. The two abilities are completely separate from each other despite the similarity in the names.
However, note the last line of Improved Divine Smite which states that Divine Smite and Improved Divine Smite can be used together and thus the extra damage against fiends and undead can be added to Improved Divine Smite albeit at the cost of a spell slot.
As of the 2018 PHB errata, the Improved Divine Smite feature description says:
By 11th level, you are so suffused with righteous might that all your melee weapon strikes carry divine power with them. Whenever you hit a creature with a melee weapon, the creature takes an extra 1d8 radiant damage.
The feature does what it says it does: 1d8 extra radiant damage on every hit with a melee weapon. Nothing more, nothing less. It has no real connection to how Divine Smite works other than the source of the power/effect in D&D lore.
(The former last sentence of the IDS description was simply meant to clarify that the damage from Improved Divine Smite does not count against the Divine Smite feature's 5d8 maximum damage limit; however, its phrasing caused people to mistakenly assume the opposite, so the line was removed. And the max damage limit of regular Divine Smite goes up to 6d8 against undead or fiends.)
In short: Improved Divine Smite doesn't do any additional extra damage to fiends or undead on its own beyond the regular 1d8 that it does to every other creature. You'd have to use Divine Smite for that.
