The numbers add up perfectly well.
DPS = total damage per cycle / time per cycle (in seconds)
Total damage per cycle = total melee damage + weaponskill damage
Total melee damage = Melee rounds per weaponskill cycle * damage per melee round
Set 1:
Melee rounds = 8.08
Damage per round = 608
Total melee damage = 8.08 * 608 = 4,912.64
Weaponskill damage = 2482
Total damage per cycle = 2482 + 4912.64 = 7395
Total cycle time = 1910 delay = 31.833 seconds
DPS = 7395 / 31.833 = 232.3
Set 2:
Melee rounds = 8.24
Damage per round = 587
Total melee damage = 8.24 * 587 = 4,836.88
Weaponskill damage = 2210
Total damage per cycle = 2210 + 4836.88 = 7047
Total cycle time = 1781 delay = 29.683 seconds
DPS = 7047 / 29.683 = 237.4
The issue is that in set 1, 8.08 rounds + weaponskill took 1910 delay, while in set 2, 8.24 rounds + weaponskill took 1781 delay. Factoring out the weaponskill delay, set 1 was at 221.5 delay per round, while set 2 was at 201.6 delay per round, 9% faster than set 1. Being that much faster, of course set 2 is going to have a higher DPS than set 1, as it more than compensates for the slightly higher damage per round (3.5%) and per weaponskill (12%) that set 1 has.
DPS is 'Damage per Second', which means both damage done -and- time it takes to do that damage matter in getting a final number.
Since you mentioned checking the head slot, I'm going to take a wild guess that you were comparing the AF3+2 head with something like the Uk'uxkaj head, looking only at the haste values, but forgetting about the DW that the AF3 head adds. That would also explain the increased number of rounds in set 2, since the extra DW would lower TP per hit, requiring more hits per weaponskill.
Uk'uxkaj will certainly do more damage per hit and per round, but the AF3+2 head will be far faster. Unless you're hitting the delay cap, you'd gain more from the AF3+2 hat.
Also, just realized I'm making an assumption about this being for nin. Only saw the single reference to that job in the first post.