Each veterancy level increases the accuracy of the unit, therefore increasing DPS. The first few shots are always inacurate, but as you stay in combat your shots become more and more accurate, until they reach their maximum accuracy, with the following DPS values:
8 Tile Range (Max):
Rank 0: 4
Rank 1: 7
Rank 2: 10
Rank 3: 18
6 Tile Range
Rank 0: 8
Rank 1: 10
Rank 2: 16
Rank 3: 21
4 Tile Range
Rank 0: 14
Rank 1: 17
Rank 2: 20
Rank 3: 21
2 Tile Range
Rank 0: 18
Rank 1: 19
Rank 2: 20
Rank 3: 21
1 Tile Range (Keep in mind shotguns are at play by this point which helps a LOT)
Rank 0: 19
Rank 1: 21
Rank 2: 22
Rank 3: 23
At rank 0, a lot of your shots miss. This doesn't mean they are useless though. Many times, the shot that misses an enemy hits another nearby enemy, therefore the real difference is smaller than what is written here. Don't forget that bugs swarm, and most of the times they have friends nearby. For a tanker or royal guard, this DPS difference could decide between squirming out a win, or losing your unit.

These guys gain extra range as they level up. Their max range in tiles:
Rank 0: 13
Rank 1: 14
Rank 2: 15
Rank 3: 16

These guys get a RoF increase and/or extra range. These guys benefit greatly from leveling up.
Rank 0: ~4 sec reload time, 8 Range
Rank 1: ~4 sec reload time, 9 Range
Rank 2: ~3.5sec reload time, 9 Range
Rank 3: ~3 sec reload time, 10 Range

As far as I see the RoF is the same, so a minor DPS increase is this unit’s reward:
Rank 0: 17
Rank 1: 18
Rank 2: 19
Rank 3: 20

It is an accuracy increase like normal rifleman
6 Tile Range (Max):
Rank 0: 13
Rank 1: 14
Rank 2: 15
Rank 3: 16
4-0 Tile Range:
Rank 0: 14
Rank 1: 14
Rank 2: 15
Rank 3: 16

They seem to get a DPS increase
Rank 0: 48
Rank 1: 49
Rank 2: 52
Rank 3: 56