Why do you kill the large jungle monster first?

If you smite it you remove a larger portion of incoming damage, and also not smithing the biggest source of gold/buffs leaves it open to stealing by the enemy jungler.

Because the "big" monster of each camp does the most damage by far, and removing it should be your top priority to protect it from an enemy invade. Smiting means that you can chunk it down near the end, securing the gold and buff for yourself.

Comparitively, by the time you kill the little mobs, you have let the big one wail on you unopposed, getting in several free hits. Now you have let yourself take damage and time your particular choice may not have. Even IF you save smite for the big one, the free damage it got on you easily outweighs it's usefulness at that stage.

Because you don't need to kill the smaller mobs as much and the longer you are fooling around, the more likely you are to be screwed by an early invade.

Because the little ones do practically zero damage. Their only purpose is to block respawns and pad out how long it takes to clear things. Also the big one is worth money, and it's important that it dies before anything could go wrong.

Agree with what others have said.

Another point. If you're a spirit of the ancient golem jungler, you clear the big creeps and leave one little creep hitting you. This procs the golem passive which essentially gives you a blue buff for mana regen, and a huge boost in hp regen. You can sit there healing while you wait for a gank to open up. Not as effective if you leave the big creep since it does more damage and the enemy team might steal it.

When it comes to golems i kill the little first cause i can get rid of it faster and it hits almost as hard as the big one.