Before doing any of the below, my thoughts were: "Almost certainly Paladin, simply because the Paladin has ways of increasing their damage while the Barbarian doesn't."
Assuming each combatant is level 5 with 18 Strength, and the Barb has 16 Con while the Paladin has 14 Con:
Raging Barbarian: Hits and deals 2d6+4+2 = 13 damage.
- Using Frenzied Strike & Extra Attack, they hit 3 times for 39 damage.
Paladin: Hits and deals (1d8+3 or 4)/2=4 damage plus 2d8 = 9 radiant damage for a total of 13 damage (accounting for Barbarian damage res).
- Using Extra Attack and 1st level smites, they hit 2 times for 26 damage.
A level 5 Barbarian will have 12+4*(7+3)=52 HP and ~16 AC.
A level 5 Paladin will have 10+4*(6+2)=42 HP and 21 AC (Plate Armor and Defensive Fighting Style).
The Barbarian probably wants to attack recklessly. With a +7 attack bonus, the base chance is 35%, so has a (1-0.65^2)=58% chance to hit.
The Paladin will thus also attack at advantage, so with a similar +7 attack bonus = base chance of 60%, will hit (1-0.4^2)=84% chance to hit.
Thus, the Paladin using 1st level Smites deals 0.84*26 = 21.8 dpr
The Barbarian using Reckless Frenzy deals 0.58*39 = 22.6 dpr
HOWEVER, the Paladin can also make use of BA spells. Let's say they use Branding Smite, meaning every turn they deal an extra 2d6=7 radiant damage.
- New Paladin dpr is 0.84*(26+7) = 27.7 dpr
The Paladin kills the Barbarian in 52/27.7 = 1.88 rounds
The Barbarian kills the Paladin in 42/22.6 = 1.85 rounds.
ALTERNATIVELY, the Paladin could buff their AC via Shield of Faith, giving them an effective +1.5 AC (roughly accounting for lost concentration and/or going second) and deal an extra 1d8 damage per turn via 2nd level Smites
- New Paladin dpr is 0.84*(26+4.5)=25.6 dpr -> killing barbarian in 2.03 rounds
- New Barbarian dpr is 0.51*39 = 19.9 dpr -> killing paladin in 2.11 rounds.
tl:dr. It's basically a tie, likely coming down mostly to "who goes first" and the randomness of the d20.
