I'm not entirely sure what you are asking. If you're asking "what's the difference between a shotgun optimized for sport and a shotgun optimized for military / police / self-defense use?" then the answer is configuration mostly. Shotguns are pretty crude instruments so they don't vary much in terms of actual mechanism between use cases. The most obvious difference would be ammunition capacity. Sport shotguns (depending on the sport and exceptions apply) typically carry between 2 and 4 shells because trap / skeet only require you to fire 2 successive shots at any given time. "Combat" shotguns have much higher ammunition capacities so you aren't reloading as frequently.

Additionally sport shotguns typically have long barrels (I believe because pushing the sight farther away from the shooter's eye has been shown to improve accuracy among other things), are heavy because weight is less of a concern, and typically lack accessory mounting points (eg for ammunition holders, flash lights, other shit) because they are unnecessary and throw off the balance of the gun which may reduce accuracy. Competition guns are also frequently configured to fire a different type of ammunition which produces less recoil for the user and puts less strain on the shoulder over long bouts of practice. This ammo may do less damage to what it hits, so it would be less appropriate if you are trying to kill a person or a large game animal.

If you are asking "what's the difference between any gun optimized for sport vs one optimized for killing things?" then what you're effectively asking is "what's the difference between a computer optimized for hitting an overclocking record vs a server optimized for running your mission critical thing in production?" Nuanced rifle differences are out of my wheel house as I don't shoot rifles much, but the biggest difference from a design philosophy perspective is the performance:reliability trade off. If your rifle fails in a competition setting, that sucks but you're not going to die. If your rifle fails in combat while someone is shooting at you, you have real problems. Like servers, rifles designed for military or self defense use are therefore designed to operate correctly under a much wider range of conditions because they can't fail, and they sacrifice accuracy to meet that requirement. The most obvious example would be the AK-47 which is notoriously reliable to the point where it's a meme due to its simple mechanism and loose tolerances, and not nearly as accurate in standard configuration as many other rifles, even other rifles used by armed forces.

The other obvious difference is fire modes in a military setting (this is obviously context dependent). There is no need for burst fire (one trigger pull firing multiple bullets) or full auto fire (holding down the trigger yields continuous fire) in a competition setting because those fire modes are mostly used for scaring your enemy and getting them to stop shooting at you. You're not interested as much in hitting them, just suppressing them. You don't need to suppress a paper target because a paper target isn't shooting back at you.

If I look at your question literally, it's a hard question to answer because anything can be used to kill someone. What's the difference between a tank and a truck that a terrorist uses to run over people?