Why would I want a hard-shell suitcase (for air travel)?

Security would be one reason, as can be seen on this forum thread when an unscrupulous airport employee tried real hard to get inside the bag:

I do not think there's any softshell that would've protected its contents against such an attempt.

Another reason would be caring for the bag itself. One my bags once developed a tear from what was clearly dragging on oily tarmac. I was fortunate enough to make it home with it and Eagle Creek fixed it (for free, because EC has a warranty like that, that's why we buy EC) but just as well the bag might have given it up completely. There are horror stories of other suitcases or bent conveyor parts acting like an knife. It takes a lot more to tear my Samsonite Firelite apart than a softshell -- even if it's made of such sturdy materials like 1000 denier Cordura fabric that Red Oxx uses. In the past I had a Pelican 1620 which probably could have withstood anything short of a direct nuclear strike (the photo is a Pelican 1514) but I have parted with it because I can't justify the weight vs sturdiness ratio and now just check the Firelite in. It's sturdy enough for me.

The advantage or lack there of is totally dependent on your packing practices. What do you take when you travel and how full is your luggage each trip.

If you take clothes and a simple toilet kit, then there is really no added advantage to a hard sided suitcase, other than perhaps some waterproofing if the cloth sided suitcase is not coated nylon.

On the other hand if you take fragile bottles of perfume or a few bottles of whiskey to drink yourself to sleep in the room, then then the added protection of a hard sided suitcase comes in handy.

From a fullness point of view, a hard sided suitcase is subject to items being thrown around during handling if it is not filled completely with clothing and such. But whether this matters or not again depends on what you pack and whether or not you compartmentalize your packing using things like Packing Cubes.

Ultimately there is no singular answer. You need to consider what you take on your travels, what souvenirs you might buy to bring home and how fragile all these are. Then buy a bag that works best for your specific needs, not what the advertisers claim.

Protection for both things inside and people outside.

Since I have changed to exclusively getting hard-sided suitcases I have never had any item inside the suitcase get damaged. With soft-side ones, every one out of five or so flights, something would get at least a little damaged or bottles squeezed out.

The other reason to buy a hard-sided case is to make it harder for handlers to get at your stuff. There is a difference, although that does not mean they won't be people who still get it. Note that you must look at suitcases specifically for that purpose if that is important to you. If the suitcase has zippers or built-in TSA-Locks, it won't be hard opening.

Not only can it have protective value for fragile stuff, but it can have protective value for stuff that won't appreciate being crushed. I owned one hardside (unfortunately, it was a spinner and the wheels didn't survive it's last flight) for transporting packets of powder. The only protection needed is to keep them secure against moving around. Anything the container will survive they'll also survive. However, they do not fare well with a container that can deform. When I transported some in a softside bag multiple packets failed. My first try in a hardside had a couple of failures where I didn't adequately secure them against movement. Secured against movement, no damage even when the airline smashed the wheel.

Another angle then protection:

Compliance with regulation and optimization.

I fly often on trips where I only have carry on luggage. I have a hard-shell suitcase that's exactly the maximum dimensions allowed to bring as carry on. This way I can cram the maximum amount of stuff within the allowed dimensions.

If you try the same with a soft bag, it will start to lose shape and you run the risk that "it won't fit" and you have to pay extra fees.

There are two kinds of hard case suitcases being sold, or at least named. The lightweight and the more massive non lightweight (?) ones.

The lightweight kind is somewhat of an hybrid- it's not nylon but still have some elasticity to it. It's not easily punched, but it's not bullet proof.

I have experience with two kinds of Delsey cases, the lightweight one survived for less than a year- it fell on its corner during delivery and was punched, this is something that would never happen in other kinds. The "real" hard one survived for years, but was much more heavier.