I would really like to address this.
An extra +1 to hit and +1 damage at first level has some effect. Your group of two fighters is engaged with a darkmantle. AC 11, HP 22. The difference between a +4 to hit and a +5 to hit is the difference between rolling a 7 versus a 6. We will say the combat lasts four rounds. (Run simulator) In ten different, four round fights against a darkmantle, the amount of times the +5 fighter outpaces the +4 fighter is 2. So there are two rounds that the +5 fighter hits and the +4 fighter doesn't. In 40 rounds!
The difference in damage is even more minimal. As the +5 fighter will have done approximately 2 more points of damage than the +4 fighter. In 40 rounds! 2 points - 40 rounds.
And if we move them up to level 4 and have one take a feat and the other doesn't, well, there is no math for that. And that is point, everyone argues that the +1 is such a big deal. It isn't.
What it does do is it creates a clear path for people to see what they think is "better." For others, it creates a path for them to see beyond the +1, and focus on something different.
EDIT: I would like to add this doesn't even take into account initiative, which is dex based. So if your +4 fighter was a halfling, odds are they might be faster than the +5 fighter, thus getting to go first, and quite possible ending the round on their turn.
This is
not even possibly correct, mathematically. I don't know what simulator you're using, but as a simple matter of math, there is zero possibility, that over 40 rounds, and presumably at least 40 attacks, even if some of them miss, a +1 damage
on every hit translates to just 2 more damage.
You've screwed that up in some way. Or the simulator is not working right. +1 damage on every hit, dude, and hitting more often. If you point me to the simulator I can probably spot your error. Also, you say there are two rounds where the +5 Fighter hits and the +4 one doesn't, yet you're claiming only 2 more damage? Do you not see the obvious problem there? They'd have to have hit for only 1 damage each time - and to have done 0 damage more on every round where both hit. I mean what? Come on dude - this is why they still try to teach kids to do basic maths without a calculator - you need to be able to recognise when you've got an obviously-wrong result because your input was in error.
(One obvious error here could be you somehow set them to doing exactly 1 damage each on a hit - that would result in what you're describing.)
The only non-error possibility I see is if it's a weird mathematical artifact of choosing a Darkmantle specifically (low AC, high HP) resulting in some kind of overkill - this would be especially likely if you were using a fixed damage value. If so though, that's still an error on your part, just in the failing to think it through or understand the maths part, rather than an actual error-error.
I mean, just looking at a straightforward DPR calculator, for 1 attack and AC11, I'm seeing 7.35 DPR for +5, and 6.4 DPR for +4, which makes sense. Over 40 rounds that should mean the +5 Fighter does 38 points more damage.
As soon as you get an extra attack, the DPR difference is doubled, too - and that's level 5. Advantage also helps the +5 Fighter more, because it means he's more likely to land an attack when the +4 guy misses (please don't make me get into the detail maths here, it should be obvious to you why this is).
I find the idea that there's something offensive about modelling the difference between a 3ft Halfling and a 8ft Goliath or Minotaur somewhat absurd.
Good thing no-one has suggested that then, eh?
It's not offensive, but it is a fool's errand to try and "model" that sort of thing in D&D combat-relevant rules, given how poorly D&D models everything and how abstract it is. It's actually bad game design if you put your "modelling" ahead of the game working well.
Weirdly 5E does have a good way to model this difference which doesn't cause combat problems, too - Powerful Build. Even if you don't track encumbrance this can come up a lot for dragging things, carrying people around and lifting unreasonable objects (in my experience it does). The vast majority of races which are either "big bois" or have +STR get it. But randomly some don't - Dragonborn and Minotaurs do not, for example, despite being larger than some races who do and having +2 STR (Orcs, Hobgoblins).
(Weirdly 5E PC Minotaurs are quite small - they average 6'1" and 238lbs. That means I'm bigger than the average PC Minotaur, which troubles me. That's a whole other discussion though.)
Also Powerful Build more reliably models greater strength due to size/leverage than +1 STR mod does even. Right now, if you make, say, lifing a 1000lb portcullis into a STR or Athletics check, a Halfling with +3 STR mod and a Goliath with +4 STR mod can, RAW, do it almost equally well (I think most DMs would question how FAR the halfling could lift it lol but that's beside the point), with just the obvious 5% chance difference. Whereas if you look at the weight and compare it to max lift, then the Goliath with STR 18 should be able to auto-lift it, as they can lift 1080lbs thanks to Powerful Build. But the STR 16 Halfling who is both size S and doesn't have Powerful Build can only lift 240lbs. Even if the Halfling was STR 18, he could only lift 270lbs.
That, to me, is a much better way to "model" strength/size differences. And it's already in the game...