You played against assassins, that's what happened.
IF you do not respect assassins, you die over and over and over and over again.
And no, Akali doesn't heal more than Vladimir.
Akali:
Twin Disciplines STATIC COOLDOWN: 8 / 7 / 6 / 5 / 4
Twin Disciplines
Akali's first two basic attacks are empowered.
DISCIPLINE OF MIGHT: Akali's first basic attack heals her for 5 - 60 (based on level) (+ 60% bonus AD) (+ 45% AP) health.
DISCIPLINE OF FORCE: Akali's second basic attack within 4 seconds deals an additional 10 - 100 (based on level) (+ 50% bonus AD) (+ 75% AP) on-hit magic damage.
At lv 6, akali heals every basic attack with a cooldown of 6-5 seconds for 15 + 60% bonus ad (She usually has none) and +45% ap, she usually has some of this, lets say she has 100 ap for whatever reason at lv6. She heals for 60hp every auto attack every 5 seconds.
Vlad:
Transfusion TARGET RANGE: 600 COOLDOWN: 9 / 8 / 7 / 6 / 5
Transfusion
ACTIVE: Vladimir drains the lifeforce of the target enemy, dealing magic damage and Redemption item healing himself.
MAGIC DAMAGE:
80 / 100 / 120 / 140 / 160 (+ 60% AP)
HEAL:
20 / 25 / 30 / 35 / 40 (+ 35% AP)
Each cast of Transfusion generates a stack of Bloodthirst over its cooldown, which is displayed in his secondary resource bar. At 2 stacks, Vladimir gains Crimson Rush for 2.5 seconds and 10% bonus movement speed for 0.5 seconds.
So, lets say Vlad also got 100 ap at lv6.
He heals for 30 + 35 every 7 seconds. From range. So, that's 65 heal (over akali's 60).
And CRIMSON RUSH: Transfusion deals 85% increased damage and heals for an additional 30 - 200 (based on level) (+ 5% (+ 4% per 100 AP) missing health), reduced to 35% against minions.
And once every 3qs he heals for an extra 80+5% of missing hp + 4% per 100ap of missing hp.
So, while Akali has smaller cooldown on her passive heal.
3 auto attacks from akali that heal her would heal for about 200hp and would require 18 seconds.
Vladimir on the other hand will heal in 21 seconds a total of:
65+65+80+9% missing hp. So, basically 210hp + 9% missing hp.