What offers better protection?

The diamond chestplate is better.

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So, how armour and enchantments work is that first the armour reduces the damage you take, then the enchantments reduce the remaining damage. (This is further complicated by the fact that the exact amount of damage reduction an enchantment does is slightly randomised each time you're hit, but it varies evenly between a 1x and 1/2x protection multiplier, so we can just average that to 75%.)

So we can have a neat formula:
total reduction % = armour % + (avg. enchant % × (100 - armour %))

A diamond chestplate reduces 32% of incoming damage (4% per half-shirt in the HUD, and a diamond chestplate gives 4 shirts or 8×4%). Protection I reduces the remaining damage by (on average) 3%. The total reduction (on average) from a Protection I diamond chestplate is therefore 32% + (3% × (100 - 32)) = 32% + ~2% = ~34% damage reduction.

An iron chestplate reduces 24% of incoming damage (4% per half-shirt, of which it gives 3). Protection III reduces the remaining damage by (on average) 9%. The total reduction is then 24% + (9% × (100 - 24)) = 24% + ~7 = ~31%.

So the Prot I diamond chestplate wins out by 3 percentage points of damage reduction.

However, there is a further wrinkle. This only applies to wearing just the chestplate. If you're wearing other armour, the enchantment on the chestplate kicks in after all those other bits of armour also reduce the initial damage, so an enchantment on one piece of equipment is more effective when you wear it with other, unenchanted armour.

Let's assume you're kitted out with non-magical iron gear, except for these two chestplates.

Iron boots, helmet, leggings, and a diamond chestplate reduce the incoming damage by a total of 68%. That means with Protection I (3% reduction of remaining damage) the total is 68% + (3% × (100 - 68)) = 68% + ~1% = ~69%.

Full iron gear is a total damage reduction of 60%. With Protection III (9% of remainder), that's a total of 60 + (9% × (100 - 60)) = 60% + ~4% = ~64%.

So when you've got other armour, the Protection I diamond chestplate widens the gap, beating the Protection III iron chestplate by 5 percentage points.

It definitely depends.

1V1 FIGHT

Diamond is better by just a little bit. However, if you enchant your iron armor one more time to Protection IV, then the iron armor is actually better than the diamond by 6 hits!

TNT/CREEPER EXPLOSION

Ok, the answer’s obvious here. The diamond armor is better than the iron armor already. (TNT can one shot FULL iron armor depending where you stand.) Enchanting the diamond are will just make it even better (Still, might be taken down in 2 explosions depending on where you stand.)

So who’s the winner overall?

In a 1v1 fight with bows, swords, and TNT, it’s really a close tie. But in the end, believe or not, I think iron would win. TNT takes a hot moment to blow up, which means the iron armored player could escape easily, and win the fight. Well, depending how pro the player is...

P.S. I may be TOTALLY wrong here. Also, sorry I’m late.