The difference in Damage on a single target from the abilities alone is a matter of usually less than 10 points of damage. The Ranger will always outdo the Huntress for DPS with everything else the same because the damage scaling applies not only to abilities, but to every shot they fire.
The Ranger's Piercing Spreadshot does offer more DPS on single large targets like Ogres and bosses because of damage scaling, not because of the "shotgun" effect of hitting with all 3 projectiles because Piercing Shot and Piercing Spreadshot are essentially the same in terms of calculation of damage dealt yet Piercing Spreadshot uses a slightly lower value for each projectile. If the Ranger and Huntress had equivalent damage scaling, Piercing Spreadshot would do the same or usually less damage than Piercing Shot by up to 4 points of damage due to rounding, but that's not a problem because they don't have equivalent damage scaling.
It is important to realize that damage scaling calculation for the Ranger ramps slightly higher than that of the Huntress and so the Ranger will always do slightly more normal damage with everything else the same by several points of damage. Because of this, the weapon damage being fed into the formula for Piercing Spreadshot has a slightly higher value than that being fed into Piercing Shot by the Huntress and so it will always do more damage in total.
As Gwen stated, the recharge time for Piercing Shot can easily be seen to be 4.0 seconds in the property inspector for Ability_Huntress_PiercingShot listed as Activation Interval. The recharge time for Piercing Spreadshot has been observed to be 4.0 seconds. To test it yourself, quickly hit the hotkey for Piercing Spreadshot repeatedly and see that you are told that the cooldown remaining is 4 seconds and, as the game rounds the cooldown time in the message up to the nearest integer, the answer is right there.
The Mana cost is 40 for Piercing Shot and 50 for Piercing Spreadshot.
It's also worth mentioning that in addition to these abilities' attacks, they also fire a single shot of your weapon with its normal properties without costing a shot from your weapon's supply, meaning that they also provide essentially a free shot that you won't need to reload. Due to damage ramping, this also further spreads the damage for single targets DPS only.
Details for Piercing Shot are easy to come by as the details of this ability can all be inspected in the Unreal Editor for Dungeon Defenders and by looking at relevant source unreal script files. As of the current version of the tools, necessary files for the packages required to load in the New Hero DLC package were not included with the game in any version and, as such, concrete values cannot be determined at this time with regard to Piercing Spreadshot, but estimates with a little bit of math can get fairly close.
The formula for projectile damage (as seen in DunDefProjectile.uc as applied with the boolean parameters of Piercing shot (ScaleHeroDamage and MultiplyProjectileDamageByWeaponDamage as seen in the properties for Projectile_PiercingShot):
Damage = ProjectileDamage*WeaponTotalDamage*ScaleDamageStat^ScaleDamageStatExponent
Note that ScaleDamageStat is not the actual stat, but a scaled version of the stat (as graphed here using the formula in the source files and values from the game data). Filling in the values found in DunDefHero.uc for the Ability Two formula as that is what Piercing Shot and Piercing Spreadshot use, we get:
ScaleDamageStat = 1 + (0.66*((MIN(StatAbilityTwo + 1,4)^0.0825) - 1)) + (0.75*(((StatAbilityTwo+1)^0.3375) - 1))
Note also that WeaponTotalDamage is not the damage stat of the weapon, but the amount of damage that will be done with the weapon as a result of weapon damage multiplied by the scaled hero damage. Simply fire the weapon normally and the resultant damage is the value we are looking for here. Hero damage scaling uses different values for each hero. Because of different damage scaling, Ranger will always have higher values for weapon damage than Huntress.
From the same properties on Projectile_PiercingShot seen in the DDK, we can see that ProjectileDamage = 1.36, ScaleDamageStat is the stat for PiercingShot and ScaleDamageExponent = 0.93 so we get:
PiercingShotDamage = 1.36*WeaponTotalDamage*ScaleDamageStat^0.93
I haven't found exactly where in the source this happens, but it appears that the damage for Piercing Shot is rounded up. So with an effective weapon damage of 49 (through whatever combination of weapon/hero attack you like) and 0 points in Piercing Shot, Piercing Shot will do a weak 67 damage (1.36*49*1). With 5 points in Piercing Shot in stead (using the scaling formula above to get a ScaleDamageStat of 1.70302264...), Piercing Shot will do 110 damage (67*1.70302264...^0.93) which I tested and confirmed to be correct.
Assuming the same standard projectile formula applies for PiercingSpreadShot with the same boolean parameters, considering the total damage of each of its 3 projectiles, we get:
PiercingSpreadshotDamage = 3*(ProjectileDamage*WeaponTotalDamage*ScaleDamageStat^ScaleDamageStatExponent)
As ProjectileDamage and ScaleDamageStatExponent are constants, by changing the values that we can control, we can determine what they are and then compare the damage formulas of the two abilities. I did a number of simple tests to gather data and extrapolate the values. It should be noted that the formula is calculated separately and has its own rounding applied.
With 0 points in Piercing Spreadshot, firing a shot and then firing Piercing Spreadshot, I was able to determine ProjectileDamage. As you might expect, it is close to 1.36/3, with a value somewhere around 0.445 (+/-.002).
By testing with the same weapon and changing the Piercing Spreadshot stat through different equipment, I was able to determine ScaleDamageStatExponent. As you might expect, it is 0.93. This gives us the formula:
PiercingSpreadshotDamage ~= 3*0.445*WeaponTotalDamage*ScaleDamageStat^0.93
So with an effective weapon damage of 49 as before and 0 points in Piercing Spreadshot, Piercing Spreadshot will do a weak 66 damage (0.445*49*1*3 = 22*3). With 5 points in Piercing Spreadshot in stead, Piercing Spreadhot will do 108 damage (3*22*1.70302264...^0.93 = 3*36) which I tested and confirmed to be correct. However, using a weapon that had 49 damage with the Huntress, given all things the same, it would do 51 damage with the Ranger, resulting in 114 damage with 5 PSS and what should be 69 damage with 0 PSS.