Almost nothing. That choice basically is the new replacement for the old attributes in the previous games.
The developers said that the old attributes were basically just a fancy way to raise or lower Health, Magicka or Stamina (besides also influencing skills, obviously), so they went for this "choose between the three" plus "perks" system.
The only thing that changes is the amount you can carry. As Arkvive pointed out, choosing stamina will increase your maximum carry capacity by 5.
As you noticed, Destruction is pretty straightforward. Each cast and while continued casting, you will increase skill, but this has a caveat: only if you are attacking someone. If you're fizzling your spell in the middle of town, it's not going to increase it.
Alchemy is also straightforward: each potion created will increase your skill. As dpatchery notes, eating ingredients for effect determination will also increase your skill by a small amount.
With Enchanting, each enchant (or disenchant, as LessPop_MoreFizz notes) as will increase skill. dpatchery also notes that you can increase your skill by recharging items with soul gems.
Restoration, namely the healing spells, only increase skill if your health is below its maximum. The others only work if you're in combat or near hostile enemies: this includes spells like Steadfast Ward.
For summoning spells (essentially most things in the Conjuration school), you won't get a skill up for the summon until you enter the range of hostile mobs and, in the case of summoned creatures (like the Atronach and the Familiar), they do some damage. With bound weapons, entering range of hostile mobs is all that's necessary to gain Conjuration skill; however, using them in combat will increase their appropriate weapon skill, not Conjuration.
Alteration and Illusion spells that are targeted towards messing with hostile mobs need hostile mobs within range to skill up. Buffs, like Oakflesh, don't increase skill without hostility as well. Others, like Candlelight and Muffle, are recastable without hostility but only increase skill by a small amount.
Best Answer
If you plan on using Illusion, primarily, that's a once or twice per battle kind of magic. You're not spamming illusion spells nonstop in an effort to kill everyone. You disable the big threat and cause some chaos, then add in damage with your bow at range.
Being a stealth assassin type, you also shouldn't (theoretically) be taking a lot of damage. Your companion (which you should use) should be acting as a meat shield while you put some arrows in the target, and only really beefy opponents in really narrow passageways should be damaging you. Especially if you use illusion's calm/fear effects, your health should be pretty safe.
That being said, things never go fully according to plan, so health is still a good choice. As an archer, you need stamina to zoom and slow time, and you'd be surprised how quickly it runs out once you get those perks. I have built my archery/sneak/illusion assassin using a 2:1 ratio putting points into health:stamina, and never putting anything into magicka. At some point I would like to increase my magicka by about 50 points, because it seems a little low, but at level 30 so far this seems to be a good mix.