Damage per second is the standard way to compare weapons, and for a long fight against a large health pool it is the right one. Over thirty seconds of sustained fire the arithmetic smooths out and the weapon doing more damage per second does more damage.
A duel is not that fight. A duel lasts four or five shots, and over four or five shots nothing smooths out at all.
Damage arrives in lumps
Against a target with 150 health, a weapon doing 25 damage a shot needs six shots. A weapon doing 28 damage a shot also needs six. The second weapon has 12 per cent more DPS and kills at exactly the same moment, because you cannot fire four fifths of a bullet.
Those extra three points of damage are not slightly useful, they are worth nothing whatsoever. The time to kill calculator reports the overkill on the final shot for exactly this reason: at 25 damage the overkill is zero, which means you are sitting precisely on a threshold, and at 28 it is 18, which means you are carrying 18 points of damage that do nothing.
Where the breakpoints are
Shots to kill is health divided by damage, rounded up, so the breakpoints sit wherever that division crosses a whole number. Against 150 health the thresholds are at 25, 30, 38 and 50 damage. Between them, more damage changes nothing.
This is why patch notes that move a weapon from 28 to 30 damage cause an uproar while a change from 25 to 28 passes unnoticed. The first crosses a threshold and removes a shot. The second does not, and the weapon kills at the same speed it did before.
It also explains why health pools are balanced as carefully as weapons. Raising a character's health from 150 to 155 adds a shot to every weapon sitting exactly on 25 or 30 damage, which is a far larger change than the five points suggests.
What DPS is still for
Within a shot-count band, the only thing separating two weapons is how fast they fire. Six shots at 450 rounds per minute takes two thirds of a second; six shots at 900 takes a third. There the DPS comparison is exactly right, because the shot count is already fixed.
So the two numbers answer different questions. Use DPS to compare sustained output against something with a lot of health, and when weapons sit on the same shot count. Use time to kill when the fight is short enough that the rounding dominates, which in most shooters is nearly always.
The same lumpiness turns up wherever a continuous stat meets a discrete outcome, which is most of a game. For the stat allocation version of the problem, see the piece on crit chance and crit damage.