The Alliance wanted to reduce the reload time of the original Widow rifle, without sacrificing its stopping power. The solution was to increase the number of shots the gun could fire before it needed a fresh thermal clip. Heavy for a sniper rifle, the Black Widow's firepower more than compensates for its encumbrance.
The Black Widow does about 25% less damage per bullet than the standard Widow. However, it fires fast enough that it empties all 3 rounds in the time it takes a Widow to fire one and reload. Thus it effectively deals 125% more damage, if you ignore the reset time, accuracy penalty, animation cancellation, and quick-scoping. If this is not ignored, then the M-98 Widow does more damage per second (DPS) than the Black Widow.
Much like its earlier counterpart, the Black Widow is one of four sniper rifles in multiplayer capable of piercing cover without using the aid of Armor-Piercing or Drill Rounds, or the use of a Piercing Mod or High-Velocity Barrel. It has an innate 25cm penetration, which is the lowest of the weapons with innate penetration, capable of only piercing through Guardians' shields and other light cover. Thus, without appropriate mods and/or Ammo Bonuses, it is inadvisable to attempt firing at enemies behind cover as there's a high chance that the cover is too thick for it to pierce through. With penetration-increasing bonuses, however, the penetration distance gets larger, but that won't increase the damage done to enemies after penetrating through objects.
However, the Black Widow is better utilized as a crowd controller and anti-armor weapon. A headshot is enough to kill any unprotected troops, and against armor units (especially Ravagers and Brutes), it'll far out-DPS anything in existence.
Spare ammunition is another major factor, as you cannot carry much more than with a basic Widow. Frequent trips to resupply are needed to keep using the Black Widow. This can be countered with the Spare Thermal Clip mod, giving you 50-90% extra ammunition on reserve. More experienced snipers will always make every shot count, so an Extended Barrel may be a better option for them.
Another weakness is its high reset time after firing a shot (the time it takes to recenter the crosshairs). This makes it difficult to place follow-up shots up quickly, further reducing the potential rate of fire. This can be lessened by releasing zoom between shots.
Compared to the N7 Valiant, the Black Widow has a lower rate of fire, much slower reload, less spare ammunition, and is also much heavier. However, it has natural piercing properties and more damage per shot compared to the Valiant. This makes the Black Widow more suitable for heavy duty work (e.g. Atlases, Banshees, Scions), but less flexible for classes that are not focused on weapons training.
The recoil cancelling ability of turian characters works extremely well with this weapon, allowing more rapid firing since the rifle will not go off target as significantly.