In various wars militaries have used munitions designed to explode and spray shrapnel at enemy personnel. The shrapnel has usually either been metal darts or metal balls. An example of the former would be the beehive artillery shells used by the US during the Vietnam war and an example of the latter would be the CBU-24 cluster bomb also used by the US in Vietnam.
My question is all things being equal which is the better kind of munition for use against enemy personnel in combat?
The aim is to take people out of action. The darts (flechettes, actually), because they spin as they fly through the air, will do more damage to whomever they hit. I guess the theory is that it takes the enemy out of action for a longer period of time.
They’re used from different platforms; you’re essentially comparing apples and oranges. Flechette/Beehive/Cannister/Grapeshot are fired from direct fire weapons, and rely on the kinetic energy they receive from being fired out of the barrel. They’re all basically shotguns in that sense.
The metal balls as you describe cluster munitions are explosive weapons. Each of the metal balls in a cluster munition is a bomblet that has an explosive filler that turns the metal casings into shrapnel. In essence, a cluster munition is full of hand grenades. They don’t rely on kinetic energy from their delivery platform, even if they are fired from artillery pieces. The kinetic energy comes from the explosives in the balls turning the metal casings into shrapnel.
While @Cmyers1980’s example of a “ball” is in fact a submunition, there are weapons that use balls as direct ballistic projectiles. You named one yourself: shotguns typically use spherical projectiles (shot), not flechettes.
Also, the famous M-18 Claymore mine uses a panel of high explosives to accelerate a whole bunch of large ball bearings in a cone of death when triggered:
Internally the mine contains a layer of C-4 explosive behind a matrix of about seven hundred 1⁄8-inch-diameter (3.2 mm) steel balls set into an epoxy resin. When the M18A1 is detonated, the explosion drives the matrix forward, out of the mine at a velocity of 1,200 m/s (3,937 ft/s), at the same time breaking it into individual fragments. The steel balls are projected in a 60° fan-shaped pattern that is 2.0 metres (6.6 ft) high and 50 m (55 yd) wide at a range of 50 m (55 yd). The force of the explosion deforms the relatively soft steel balls into a shape similar to a .22 rimfire projectile.
Actually more than one 
. Cannister and grapeshot are both balls.
Cannister (British 3" Case shot):
Grapeshot:
I am not a munitions expert by any means but I would venture a guess that if your goal is affect the largest number of enemy personnel possible who are in a somewhat large area, then you go by balls that are dispersed by some sort of aerial explosive (like the HIMARS GMLRS-AW rocket with 180,000 tungsten balls).
If your goal is to inflict grievous harm on a relatively smaller number of enemy personnel who are all concentrated in a rather small area, then you go with darts.
Darts are better - if you can get them to hit the enemy point-first. If you can’t, then balls are better.
Would it really matter considering it would still be traveling as fast as a bullet?
Sure - a dart tumbling in all directions is less aerodynamic and losses more energy in flight. There’s a reason bullets are bullet-shaped.
EWAG, a tumbling flechette might be stopped by something as light as your uniform while a bullet, even if it’s tumbling, will not be stopped.
IIRC, flechette were originally looked at because, if they weren’t tumbling, they’d go through things like foliage and, more importantly, fabric armor.
FWIW the US has gone with smaller balls over other shapes and sizes as of late; the M1028 canister round for the 120mm on modern M1 Abrams uses 1,100 tungsten spheres. Page 8/17 from this General Dynamics pdf on its development shows various shapes and sizes looked at during development including darts and squares. The M1040 105mm canister round for the Stryker MGS uses 2,080 tungsten spheres; it looks like the last time flechettes were chosen for direct fire tank caliber rounds was the M494 APERS-T beehive from 1967 which used 5,000 0.8 g steel flechettes.
a relative was in artillery, & he explained a diabolical sub-munitions package delivered by shell: 100 small clamps each with a small exploding ball. But that ain’t the diabolical part - each was adjustable to either mode of explode on contact or have the clamp open ball-side up & bury itself 1" into the ground, to function as a landmine. That small ball ain’t big enough to kill some1 (from blast, anyway), but will chop up a foot. A typical load would be 1/2 set for contact, 1/2 for clamping/landmine mode.
This isn’t the exact same thing, but close:
Not a new or unusual invention at all.
A friend of mine is very pedantic on the difference between “shrapnel” and “fragments” in munitions.
I don’t recall what that difference is, but I do remember being darkly amused imagining a person dying and pleading with a medic, “Help! This shrapnel is killing me!”
“No, no - those are fragments that are killing you.”
Shrapnel was originally balls packed into a bursting artillery charge, to be violently scattered like an oversized shotgun. The casing of the artillery shell was originally thin sheet metal or the like.
As the Britannica article says, weapons designers figured out that if you make the shell casing itself thicker and design it so that it fractures reliably into regular sized fragments, you don’t have to go to all that laborious and expensive packing shrapnel balls into the shell. Just pour the explosive compound into the fragmentation casing and Bob’s your artillery victim.
I remember reading that the US military experimented with all sorts of flechette based weapons in the 70s And 80s thinking them to be the next “Wonder Weapons” and having assault rifles which fired multiple flechettes per shot. It turned out though getting flechettes to reliably go straight and not just tumble in the air was an insurmountable problem and it was all abandoned by the 90s.