I got a two handed rare sword a week ago which item power is 512, today I got a two handed sword which item power is 565.
ALL Affixes on the 512 ones have higher/better range of stats than the one that is Item Power 565, meaning if I got perfect roll on a 512 one, it would be better by miles than the other one. Almost every stat is the same type on both swords.
The only thing higher is damage per shot. Not even the weapon specific affix (the one below weapon speed stat is higher, it’s actually lower than 512 two handed sword.
So Item Power only determines Damage per Shot? (That is until you reach the Item Power breakpoint). But as long as they are both in same Item Power category, the one with much higher Item Power can roll weaker affix ranges?
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Item Power only influences certain stats like Health and base item DPS. Otherwise, the stat ranges are just based on the breakpoints.
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However.
The 565 Two-Handed Sword has worse stat range for damage to stunned enemies, it is something like [12.5 - 21.5] and the sword with 512 power has damage to stunned enemies [15.5 - 26.5].
The thresholds only change at certain Power Level breakpoints.
1 - 149
150 - 339
340 - 459
460 - 624
625 - 724
725+
If you find two items within each tier expect them to have the same ranges for rolls.
However, if that item is within 25 points (20 for rares, 25 for legendary’s) of breaking the next threshold, fully upgrading it will grant it the next tier of thresholds; this isn’t always the best choice to make but it usually is, you can still get absolutely horrible luck and the lowest rolls possible when it tiers up.
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There are named elites in the open world that drop unique rare items that have different stat rolls. Did you get the first sword in open world? Is the vendor value of it really low?
Unless it’s an ancestral I don’t think anyone would use those items, specially only having 3 stats
all items in the same item power range have the same ranges on rolls, except life and thorns. 525-624, 625-724, 725+
The only thing that increases within each of these brackets is the base armor or base damage. So it’s very important on weapons to have as high an item power as possible, but it doesn’t matter very much on armor, just as long as it’s in the highest bracket you can currently find.
This was the case on Diablo 2 and Diablo 4 as well, breakpoints have always been part of Diablo and there is nothing wrong with that. Item power on the other hand…
…
It makes perfect sense, see now we your KPI Blizzard Overlords don’t need to raise the level cap. We heard you loud and clear, no more of that instead we are doing WoW things, every season the maximum item power will increase by 10%! To the moon!
Our KPIs and psycho tests also show it’s great for player retention and short-term satisfaction! We assure you, our dear shareholders, that this is a new age, World of Diablo is upon us.
There are some good ones, the dagger is almost BiS for poison rogue. At 845 power it gives you more damage than most regular daggers, of course a 820 dagger with 4 useful stats is superior.
Just cause you have a higher item-power item does not mean it will roll high values… All of them have a range to roll in and sometimes “oohhoo bad luck, try again ?” (in the voice of the annoying gambler in Act1)
Whatever, you get the idea there 
However I wish the primary damage wasn’t as steep of a multiplier, cause that’s exactly how monster HP grows (each and every level) as well…
Wish some “secondary” bonuses were better and the primary item damage to not always explicitly grow like that… Could make some extra space for less of an impact of level scaling while also giving “breathing room” to the other options that may or may not roll in terms of damage impact 
Replying to Degi: This is just not true. These both Items I described are in same Item Power bracket, and the one that has higher Item Power has worse stat ranges. I clearly described that.
Is the lower level one upgraded fully ? I think the ranges bump up at max upgrade level regardless of thresholds.
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no, it doesn’t. You probably upgraded it which increased the values
Yea that’s ancestral before that they are garbage