What's the difference between a kitchen knife and one that can be used to kill someone? Nothing, you can kill someone with a kitchen knife—but that isn't what it was designed for. There are guns designed to kill unsuspecting people at a distance (sniper rifles), there are guns designed to be carried for self-defense, and there are guns designed for hunting or target sports. Any of them could be used to commit a murder, but that doesn't mean they are all equally suited for it. In the end a gun is just a tool. It's the intent of the person using the tool that counts.
I think you pretty much nailed the bottom of that argument. I'd like to point out one thing: from a country¹ in which firearms are heavily restricted (non-law enforcers can't carry), there is little stigma if any on recreational shooting², and it's certainly a right to own such firearms at home. You just can't take them to work or to do shopping, only at the shooting range or equivalent, with precautions like partial disassembly, unloaded, in a bag... Normal practice.
So it's really not about weapons in and of themselves that the rest of the world is puzzled about this American debate, not about the supposed cruelty of everyday people (nobody believes that). It's really about the fact that carrying a gun to work is a very slippery slope, even if the gun is in the car. Same idea with not carrying dangerous explosives if you can help it, the risk is too high compared to most perceived benefits.
I think the self defense argument is very much biased by the fact that once others have guns, you may feel threatened not carrying yourself; conversely if no one carries you'd rather it stayed that way... It's a snake eating its own tail from both sides.
The truth is, it's actually not normal people carrying that kills a lot in the US (although child accidents are statistically too high compared to eg Europe or Asia iirc). It's really the problem of gangs etc. Most lethal shootings are statistically related to someone's lifelong "job", not everyday honest people. But removing guns from wide circulation means we don't have e.g. teens shooting others anywhere else in the world, nowhere near the same magnitude, which is a troubling fact. Indeed, it's the person that holds the gun that counts, and young minds shouldn't have access to guns in that regard. Not enough control yet, it's a biological fact.
[1]: France, but it's the same culture in most western EU countries afaik. Not sure about those closest to the Russian federation but I'm inclined to think they generally agree with us on the matter.
[2]: Hunting is certainly midly popular here in rural areas, and those who voice criticism are 99% about the animal cruelty angle, they couldn't care less if the killing was done with knifes or arrows instead. The gun angle is just not a thing in most countries where guns are effectively banned from regular society but obviously totally accessible for sports: it's OK, really.
> I think the self defense argument is very much biased by the fact that once others have guns, you may feel threatened not carrying yourself; conversely if no one carries you'd rather it stayed that way...
Guns are not a prerequisite for feeling threatened. The self-defense argument is based on the fact that two arbitrary people armed with guns are much more likely to be evenly matched than two disarmed individuals. In particular, habitually violent individuals tend to be much more experienced at, and prepared for, unarmed combat than the general public. Skill with firearms also benefits from practice, of course, but almost any armed individual would at least stand a chance of winning against a determined attacker, whereas someone without extensive martial arts experience would be unlikely to successfully defend themselves in hand-to-hand combat. Guns represent an equalizing force.
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?