The two benefits are separate

The advantage clause cares whether the target has taken a turn. The automatic-critical clause specifically requires the target to be surprised.

Winning initiative is not the same as surprise

A creature can roll lower initiative than you without being surprised. Assassinate does not create surprise by itself.

Stealth and party coordination matter

If the party approaches openly or an ally alerts the enemy before the ambush, the DM may determine that the enemy is not surprised. The subclass therefore gains more from coordinated infiltration than from initiative alone.

Critical Sneak Attack

When Assassinate turns a hit into a critical hit, the normal critical-hit rule doubles the attack's damage dice. This includes Sneak Attack dice and other damage dice that are part of the hit.

Death Strike builds on the same setup

At 17th level, Death Strike adds another saving-throw-based damage multiplier against surprised creatures. Because the setup is demanding, the feature is extremely explosive when it applies but should not be assumed to apply in every encounter.