Aside from being able to interrogate an alien in the Alien Containment chamber, are there any particular advantages to risk trying to stun an alien instead of just killing it outright?
The big benefit is that instead of weapon fragments, you can capture their weapons intact.
Early in the game, when resources are scarce, this can give you an advantage - you don't have to spend money on advanced plasma weaponry, you can just steal the alien's weapons and use them against them. You can also "sell" the spares in certain cases, but only if the council requests them.
Do note that you'll have to research the weapon before you can equip it. Having one to research can bypass part of the research requirements, but you'll still have to have researched certain prerequisites before you can research plasma weapons.
For instance, you can research "Plasma Rifle" without researching "Light Plasma Rifle" if you have a Plasma Rifle weapon from a stunned enemy. In either case, you'll need to have researched "Weapon Fragments" though.
You get to capture it's gear intact this way. Alien tech is expensive to manufacture. A few captures here and there and you'll have plenty of plasma rifles (all varieties) at your disposal.