Divine Favor (from a level 1 spell slot) lasts for up to 10 rounds, and on a hit does 1d4 radiant damage.
On a critical hit it adds 2d4 radiant damage.
Divine Smite (from a level 1 spell slot) is good for 1 hit, but does 2d8 radiant damage.
If the hit was critical it does 4d8 radiant damage.
Assumption 1: Combat lasts from 3-6 rounds
a. That's how long combat tends to last with this group
b. For a two round fight divine smite is without question the better choice.
Case 1: hit Armor Class 14
Case 2: hit Armor Class 16
While I can see Divine Favor being a decent use of a level 1 spell slot once the Paladin has two attacks per Attack Action at level 5, I am trying to nail down (in tier 1 play) how many rounds the combat needs to last for Divine Favor to be the better choice for a level one spell slot.
Assumption 2: paladin makes concentration save if hit during combat.
That assumption is to keep the question within a tight scope. As I discuss this with my paladin player, a separate risk assessment needs to be made vis a vis loss of concentration.
Assumption 3: single class paladin, level 2, 3 or 4.
My back-of-the-napkin-math says "4 rounds" but I may not be setting the problem up correctly. If every attack hits, 4d4 is slightly more damage than 2d8 on average: 10 versus 9. But every attack does not hit.
How many rounds does combat have to last to make divine favor the better choice?
My instinct is that against one big monster, use divine smite to nova the monster down; against a mob of small creatures use divine favor to whittle down a lot of targets - but that is a separate question/problem.
Note: I am the DM in this case, not the player. The Paladin is Dragonborn, and does not have the Sentinel feat. Strength is 18 (Rolled stats). Paladin is sword and board. An answer for a strength 16 paladin (point buy) is acceptable.
Lastly: if you believe that the answer is "It's never the better choice, and here is why" - demonstrating the "why" answers the question. (For example, if the "why" shows that it's not better until the seventh round, it meets the question/problem constraints).