Is the essense thief 12% damage would do more dmg against enemies in pvp more than a set that gives out weapon damage like briarheart?
Briarheart do only need crits to procs and Essence Thief need a light attack which then create a purple pool that you need to move and pick up. In fights where you move a lot Essence Thief might not be as good as it could be on paper as in general you will want to avoid to go where it moves.
Here you can see video of how it looks when it procs (mind you this is static target that doesn't hunt you or prevent you from moving like in PvP [and also happens during PvE during boss fights in dungeons/trails]).
It is 10% after recent changes.
Your typical effective weapon power is calculated like (source - UESP)
EffectiveWeaponPower = (round(Stamina/10.5) + WeaponDamage)*(1 + WeaponCrit*AttackWeaponCritDamage)*(1 + CP.PhysicalDamageDone)*(1 - AttackPhysicalMitigation)*(1 + Target.DamageTaken)*(1 + DamageDone)
So Essence thief is working similar to Minor berserk and will net best results if other "damage done" sources are absent (for example if you use 2H mace or axe, and so there is no bonus to "damage done" from greatsword).
If we take typical no-CP heavy build with 27k stamina and 4k average WD, 25% crit, 50% crit damage. 7 impen on target:
(27k/10.5+4k)*(1+0.25*(0.5-0.25))=6981 effective power.
Now let's say you will use seventh legion 5th bonus:
(27k/10.5+4k+350*1.3)*(1+0.25*(0.5-0.25))=7465 effective power
And essence thief in comparison to seventh without other damage done bonuses:
(27k/10.5+4k)*(1+0.25*(0.5-0.25))=6981*1.1=7679 effective power
Given that damage done doesn't increase healing and also your essence rune may spawn god knows where, while 7th is easy to proc, it seems like 7th is clear winner. And 7th doesn't look that good after nerf... conclusion - essence thief is niche/fun set at best.
If you're just looking for a flat damage boost (and hate the world and everyone in it) maybe consider Shield Breaker as an alternative as Thief is hard to proc. You get a flat 6% but it's doubled to 12% when enemy has a shield up.
Bear in mind though that a damage increase won't up your heals. A set with a weapon damage increase will.
The real problem with Essence Thief is that in order to get the buff, you have to run to the spot on the ground. They've improved it somewhat, but I still didnt find it very intuitive to use nor applicable to most combat situations.
You're better off stacking weapon damage because it's hard to do damage if you're dead. That being said, stam took a massive hit to survivability this patch with all the nerfs to weapon damage sets and healing nerfs.
According to the calculator it's more damage for my build than Spriggan's (on a heavy build with 7th legion as the other set). But Spriggan's is always active.