One big problem with the sling build is that it cannot attack a target more than once per turn. This means are great at taking down multiple weak opponents, but suck vs a single touch opponent. While you do get more damage out of sacred weapon it only brings the damage up to that of bow, so the gain is not that significant. You are also spending feats to do what a bow normally does without feats. That means you effectively have less feats. For this build to be fully online you have to be at least 9th level and at that point you are still lacking some important feats like weapon specialization and Point Blank Mastery.
The bow build is using a stronger weapon so that has abilities the sling build needs to spend feats for. Those extra feats give you important abilities. Weapon Specialization allows you to take Point Blank Master. Loading a sling with Juggle Load does not provoke an AoO but making a ranged attack still provokes. Precise Shot eliminates the -4 penalty for firing into a melee. That makes using the sling on foes that are in melee with the other members of the party difficult.
Assuming a 9th level warpriest with a DEX of 17 and a STR of 14 the bow will give you 2 attacks at +9 one attack at +4 vs the slings single attack at +11. If you are firing into a melee the sling takes a -4 penalty to hit bringing down the bonus to +7. If you hit with the first shot of the bow it does damage twice. The bow will do 1d8+4 damage per hit, while the sling does 1d8+9. Depending on your build the details may vary, but for comparison it does not matter. The bow does not provoke AoO so is always using the bow even while in melee. Due to the extra attacks the bow gains more from any bonuses or buffs the party provides.
The bow is hands down the better option.
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IMO, it depends on whether the sling warpriest is a halfling with the Slipslinger Style feat.
A halfling warpriest using a sling will probably be more dangerous early. A halfling warpriest can take the Warslinger alternate trait, plus use the Focus Weapon bonus feat on Weapon Focus (Sling) at 1st and their normal 1st-level feat on Slipslinger Style. This gives them +1 on attack rolls (on top of the +1 size bonus) and +1 on damage rolls with slings. You can also "treat all ranged and thrown weapons that have 'sling' in their name as if they were slings for all feats and class abilities that require such weapons, as well as the warslinger racial trait;" so use a halfling sling staff (martial weapon for halflings, gain the +1 on attack rolls and damage from Weapon Focus and Slipslinger Style, reload as a free action with Warslinger) instead of a normal sling. Magic stone is also a very useful 1st-level spell for a sling wielder.
A medium-sized bow-using warpriest will probably be more dangerous at higher levels, once they can take some of the better archery feats and the scaling damage from Sacred Weapon increases a couple times.
All warpriests get weapon focus with the weapon of their choice at 1st level. In both cases the first level feat is going to be Point-Blank Shot. That is the prerequisite of most of the feats on either build and if not taken the builds fall apart. The halfling sling staff simply brings the damage to equal to that of a long bow. You still have to spend a feat to allow you to load without provoking an AoO, and even if you spend the feat you will still provoke an AoO when you attack with the sling. While you are doing that the bow build is picking up useful feats like rapid shot. This means the sling build is 1 feat behind on gaining useful combat abilities.
The build the OP proposed for the sling is even more feat intense than the archery build and does is not as good. Using the halfling sling staff will mean they can skip Sling Flail feat but that one was a trap anyways. Even with the improvements made of the build by Dragonchess Player the sling build is still weaker than the bow.
If I wanted the "most dangerous" divine archer, I'd probably lean toward elf oracle (ancient lorekeeper, Wood mystery) leveraging the Wood Bond revelation* and the FCB adding +1/6 level to one revelation (which effectively gives the same total attack bonus with bows as a full BAB class at almost every level, except for the lowest iterative, before counting divine favor or divine power); ancient lorekeeper allows the oracle to take useful spells like gravity bow, arrow eruption or flurry of snowballs or glitterdust or mirror image or sonic scream or web, etc. in place of some of their mystery spells. I'd probably also take 2-4 levels in deadeye devotee after 8th or 9th level for Imbue Arrow (and possibly some other abilities); even though the cleric list has less AoE spells than sorcerer/wizard, there are still some nice options. The character only loses one level of spellcasting progression as long as deadeye devotee levels are 4 or less, which retains 9th-level spells.
*- which explicitly works with bows
You realize that the competence bonus only goes up by +1 for every 5 levels. The FC bonus is +1/6 of level to one revelation. That means you have to take 30 level of Oracle to increase the bonus by +1. That is a complete waste of FCB. A lot of races gain +1 spell known as a FCB for Oracle. That is a lot stronger than gaining the +5 bonus at 18th level. Even the extra skill points would be a better choice. Wood Bond also only increases the bonus to hit, it does not actually increase your BAB. This means you don’t get extra attacks for a high BAB and feats that scale based on BAB are not affected.
Taking Deadeye Devote also means your revelations do not increase. That includes Wood Bond. It also slows down gaining benefits from your curse. The spell loss of the caster levels delays gaining new spell levels on a class that is already a level behind. Since the Oracle has limits on spells known they will have fewer AoE spells to use with Imbue Arrow.
Wood Bond (Ex): Your mystical bond with wood is such that your weapons become an extension of your body. You gain a +1 competence bonus on attack rolls when wielding a weapon made of or mostly consisting of wood (such as a bow, club, quarterstaff, or spear). This bonus increases by +1 at 5th level and every five levels thereafter.
Oracle (Advanced Race Guide pg. 23): Add +1/6 to the oracle's level for the purpose of determining the effects of one revelation.
Total attack bonus for oracle levels:
1st - +0 (BAB) +1 (Wood Bond) = +12nd - +1 +1 = +2
3rd - +2 +1 = +3
4th - +3 +1 = +4
5th - +3 +2 = +5
6th - +4 +2 = +6
7th - +5 +2 = +7
8th - +6/+1 +2 = +8/+3
9th - +6/+1 +3 (effective level 10 for Wood Bond) = +9/+4
10th - +7/+2 +3 = +10/+5
11th - +8/+3 +3 = +11/+6
12th - +9/+4 +3 = +12/+7
13th - +9/+4 +4 (effective level 15 for Wood Bond) = +13/+8
14th - +10/+5 +4 = +14/+9
15th - +11/+6/+1 +4 = +15/+10/+5
16th - +12/+7/+2 +4 = +16/+11/+6
17th - +12/+7/+2 +4 = +16/+11/+6 (this is the only point where the oracle lags behind a full BAB class for total bonus)
18th - +13/+8/+3 +5 (effective level 21 for Wood Bond) = +18/+13/+8
You can stop investing in FCB after 12 oracle levels; the difference between 20 effective levels and 21 effective levels at 18th is meaningless. If you take 4 levels of deadeye devotee (a full BAB PrC), then you still have a +16/+11/+6/+1 (BAB) +4 (Wood Bond) = +20/+15/+10/+5 attack bonus at 20th level; you just lag on getting the lowest iterative compared to a full BAB class. Note that Wood Bond is a competence bonus and divine favor / divine power grant luck bonuses, so the oracle can have a higher total attack bonus with bows than a full BAB class from one buff spell.