The reason is krugs give a bonus stun on every 5 attacks when you smite it.
Raptors will also give a buff after you smite it allowing you to see wards that see you, a temporarily oracles.
If you smite red you get a small health bonus from it.
The raptor bonus has way more potential to help when you gank a lane to see if there is any wards you are seen by when heading towards a lane.
The krugs bonus helps you take even less damage every time you proc the stun way better then a one time health bonus.
gaming.stackexchange.com/users/88592/Greatwon
It's faster to walk from Krugs, to Red, to Raptor than to go from Krugs to Raptor to Red.
Additionally, the restored health from smiting Red buff will most likely only provide much of a benefit in the event of a close fight; while with ward knowledge/clearing from Raptor camp, you can make an ambush more likely to succeed, or fail quickly so that you can return to clearing.
gaming.stackexchange.com/users/43683/Sconibulus
You don't use smite on red buff in general because most of the viable junglers (not speaking about meta) can easily survive the first jungle route.
Even with tank junglers and slow jungle clear
you will not die as long as you kite the buffs a little bit and play with good runes (depending on your champion).
It also depends on your current division.
If you are playing in low-elo like bronze to low gold you can use smite for the red buff to restore hp because you don't necessarily need the buff of the raptors which reveals stealth wards, since players dont often place deep wards early and even if they do, their map awareness is usually not too good.
I would still advice to smite the krugs.
You will probably save more health points because of the stun than you would restore by smiting red.
Additionally I would not recommend to clear the raptors camp when playing as an assassin in the jungle since they hurt a lot and you can gank in the 3rd minute.
If you are low on hp after your first jungle route it always helps to kill the scuddler crab.
It gives you a lot of health when killing it
and you won't take any damage from it.
PS:
Hard CC reduces the scuddler crabs armor and magic resistance from 60 to 10 which makes it a lot easier to kill it fast.
gaming.stackexchange.com/users/119821/Timo Jokinen
Smiting Red Buff just gives a small health bonus, altough smiting raptors gives you a buff to detect and destroy a ward.
That's much better as long as you don't need more health in the jungle!
gaming.stackexchange.com/users/119259/Sime Kappa
Season 5 Jungle has seen a lot of change from its past season jungles, and most if not all champs tend to start with smiting a large monster that gives them a buff for clearing jungles.
Gromp (Blue Side Frog) gives you the ability to deal poison damage to your attackers, and Krugs (Red Side Rocks) gives you the ability to stun per 6 attacks.
Now, smiting Red and Blue buff does not give any sort of clearing buff or whatsoever.
Instead, it gives you back a small amount of HP or MP back (HP for Red, MP for Blue).
This is also true for Murk Wolves (Wolves) which gives you a ward in your Blue side jungle that chases enemy champions and Crimson Raptors (Chickens) which gives you true sight around you for a duration of time.
Why do people usually smite chickens instead of Red Buff
you might ask, this would give you a safer clear (not necessarily better).
However, the difference here is that at the end of your clear, you would want to gank a lane to pressure the lane.
In this case, the true sight is immensly powerful as it gives you the power to detect Stealth Wards.
If you happen to find any, removing them would force them to place another ward there (-75 gold for the enemy) or lose the precious vision there which often result in a devastating gank from mid or jungle.
However, this is still situational as if the enemy team jungle is shaco or nunu, you would want to smite the wolf in order to gain the ward in your side of jungle in order to prevent counter-jungling.
Or you can also grab your clearing buff (either krug or gromp) then get the Red and the Blue buff in order to prevent the enemy jungler from taking any of those.
Also, do remember to take the small river scuttler (crab) in the river - they essentially give you free vision in the river, which will allow better control over dragon / baron.
gaming.stackexchange.com/users/86613/Kim Minseo
I think it's kinda situational which start is better.
I'll sometimes save my smite for the buff depending on what the jungle champs I'm playing and the enemy is playing.
If I'm playing against a champ that like to invade early and go for a fb (like Shaco), I'll save smite for the buff because it will keep my health higher and give more time for my team to collapse if I'm invaded.
Also, if I'm playing a champ that I'm trying to hard-farm to 6, I feel the raptor buff is less useful than the heal.
However, if an early invade is unlikely and I want to go for a gank after clearing one side of the jungle, the raptor buff is hugely beneficial.
gaming.stackexchange.com/users/42464/GeneralMike
It depends on the champion you're playing and what your plan is for the early game.
If you have decent sustain then getting the instant health from red buff could be less useful than the raptor buff (which gives early vision control) or the wolf spirit (which gives you early warning of invades).
gaming.stackexchange.com/users/118109/AverageUnknown
I usually start with Red Buff and use smite when I'm in Blue side.
But in Red Side it's better start from Smiting Krugs and Blue Buff
gaming.stackexchange.com/users/121449/Arantxa Zhao
When watching jungle guides on YouTube, people who start off with Krugs never seem to get the red buff smite bonus. Almost all of them do this.    Krugs, use smite  Red buff  Raptors, use smite   It makes more sense, to me, to do this   Krugs, use smite  Raptors  Red Buff, use smite   This way you don't rely so much on your pots because you smited back essentially two pots worth of hp.   Is there a reason for not doing this?
Why don't people smite the red buff?
gaming.stackexchange.com
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gaming.stackexchange.com/questions/230059
gaming.stackexchange.com/users/112682/Trogdor
Krugs give a bonus stun on every 5 attacks while smiting Red Buff just gives a small health bonus. Using smite on red buff can provide the player with few benefits, such as faster clear times and health regenation.
The community is providing some information regarding the use of smite.