A mouse and a controller are not two ways of doing the same thing. They report different quantities to the game, and everything about how shooters handle aiming follows from that difference.
A mouse reports movement, a stick reports intent
Moving a mouse produces a measurement of distance travelled, which the game converts directly into rotation. Stop moving and the view stops.
A thumbstick reports displacement from centre, which the game converts into a rotation speed. Hold it and the view keeps turning until it is released.
One is positional and the other is rate-based, and that distinction explains almost every difference players describe between the two.
Positional input suits precision, rate input suits tracking
Because a mouse maps distance to angle, a player can learn how far to move to land on a specific point, and repeat it without needing feedback in between.
A stick cannot do that. Reaching a target means accelerating, holding and stopping at the right moment, which is a control problem rather than a memory one.
Conversely, a stick sustains a constant turn effortlessly, while a mouse requires continuous physical movement and eventually runs out of desk.
Response curves exist to make sticks usable
Raw stick input is uncomfortable, because small movements need fine control and large ones need speed, and a single sensitivity cannot serve both.
Games apply curves that make the centre region less sensitive and the outer region far more so, along with a dead zone that ignores the smallest movements.
Tuning these is a significant part of how a controller-focused shooter feels, and it is why the same game can feel precise on one title's settings and vague on another's.
Aim assistance is a correction, not a cheat
Controller players usually receive help, most commonly a slowdown as the crosshair passes a target and a gentle pull that follows movement.
The purpose is to compensate for rate-based input being poor at fine correction, so that the skill being tested is positioning and timing rather than stick control.
The amount is tuned per game, and its strength is the entire substance of the argument about mixed-input play.
Mixed lobbies expose the mismatch
When both inputs meet, the comparison is not mouse against stick but mouse against stick plus assistance, and the balance depends entirely on how much assistance was granted.
Too little and controller players cannot compete at range. Too much and tracking assistance outperforms human hands in close combat.
Most titles now offer input-based matchmaking or let players opt out, which sidesteps a balance problem that has no clean solution.