The combat begins when the Darkmantle decides to attack. If the players don't notice the Darkmantle at the beginning of combat, they are surprised:
Any character or monster that doesn’t notice a threat is surprised.
So, thanks to False Appearance, the Darkmantle has a Surprise attack. Also, when the combat starts, the adventurers don't know that they're in a combat situation and so are not scanning the entire space all the time, unless previously stated (they are not combat-aware), so the Darkmantle remains hidden until its turn. However, all characters that are higher in the initiative order than the Darkmantle are no longer surprised at the end of their turn (or, better say, the "Surprise" property is no longer applied to them)! They are thus able to take reactions. For reference, this answer to another question (also linked in the comment of this question) breaks it down in full detail.
The Unseen Attacker advantage, however, depends on the DM's ruling, as the RAW are not specific about this.
As far as Darkmantles go, they could approach their target unnoticed. They don't have to fly, they can just drop down. One could reasonably fall unnoticed as easily as dropping a curtain on top of someone. Even if someone would notice the creature as it was falling, they'd only get enough time to look up and see the horrid array of teeth approaching swiftly.
If you still believe your Darkmantles are unable to automatically Batman your PC's, then a Stealth check is appropriate, and should be made during the Darkmantle's Move action. In either case, the Surprise Attack persists, it is only the Unseen Attacker advantage that is at stake.
Keep in mind that the target of the attack must be taken into account. If the Darkmantle attacks a character that was higher in the initiative order, the character can take reactions (like Tempest Cleric's Wrath of the Storm), regardless of whether the Darkmantle remained hidden during its movement or not. The very act of attacking reveals it:
If you are hidden—both unseen and unheard—when you make an attack, you give away your location when the attack hits or misses.
However, if the trigger for their reaction is before the attack, the Darkmantle's visibility has to be taken into account.
If the Darkmantle attacks a character that's lower in the initiative order, the character won't be able to react until the end of its own turn. So, even if his reactions trigger "on hit", he won't be able to use them. However, if the characters that went before the Darkmantle have a reaction that triggers "when an ally is attacked", they can take that reaction (again, because the attack reveals it).
There is only the final, special case to consider, where the Stealth check is only partially successful and one or more characters notice the Darkmantle as it moves in to strike. It is fully within their ability to warn the others of the threat, as they can talk out of turn (within reason). However, as it's the Darkmantle's movement that triggers the Stealth check, the players that were surprised at the moment of the attack still remain surprised. Whether the darkmantle retains its Unseen Attacker advantage or not is up to the DM and his opinion on how fast a character can react to a warning.
From a role-playing perspective, if the Darkmantle remains unseen the adventurers either didn't notice the attack until the moment it hit, or they noticed it too late to do anything about it (including removing the advantage).
Here's an example of how that could be acted out:
(A PC walks under the Darkmantle and it decides to attack.)
DM: As you explore the dark cave, you feel a slight chill up your spine... Roll Initiative!
Player: Why? Do we see anyone here?
DM: Not yet.
(Initiative order is set. )
DM (to the players at the top of the initiative order, before the Darkmantle): You are not aware of any threats and proceed as you were.
(The Darkmantle's turn comes. It makes its optional Stealth check... and remains hidden. It swoops down on its unsuspecting target and rolls its attack with advantage... success!)
DM: Alright... Aldore, your head is suddenly wrapped in a curtain of flesh and you feel piercing teeth gnawing at your cranium. You take XX damage!
(DM now describes the attacker, and the battle continues. The characters that haven't had their turns yet are still surprised. When all turns resolve, the combat continues as usual from the top of the initiative order.)
Now, if the creature missed the attack (either by lacking or despite the advantage) you could have it be because the character noticed it and dodged in the last second, or you could make the Darkmantle comically flop on the ground in the middle of the party. Your choice :)
What it means
hasn't taken a turn
means that the creature has not had a turn in the combat. That is, the assassin has beaten the creature in initiative order. Surprise is immaterial: if the creature is surprised and beats the assassin in initiative it has still "taken a turn" even though it couldn't do anything with it. It also applies outside the assassin's turn if they can somehow get a reaction before the creature's turn: if the creature moved in a way that caused an opportunity attack using a reaction before its turn the assassin would get advantage on that attack.
On your first turn ... not yet acted
In contrast the ranger cannot get the advantage out of their turn.
"Not yet acted" means not having taken an action, bonus action, movement or reaction: the reaction can occur before the creature's turn. For example in a fight between a ranger and a wizard where the ranger wins initiative and attacks first. the wizard uses a reaction to cast Shield - in addition to giving +5 to AC, it negates the ranger's advantage because the wizard has acted before the ranger's attack (Shield specifically involving some sort of mysterious time travel as it does).
Surprise does matter here because a surprised creature does not get movement, action or bonus action on its first turn - if the ranger surprises the creature they will get the benefit irrespective of if they win initiative or not (subject to the preceding paragraph).
What it really means
The nuances are so ridiculously subtle and unlikely to occur that if you want to treat them the same it will do absolutely no harm to your game.
False Appearance and Hidden are not equivalent - a creature benefiting from False Appearance is not (necessarily) Hiding. In fact, it is more common for them to be sitting in plain sight.
You can, of course, decide that the creature wants to Hide when it becomes aware of the party in which case:
See How often during combat can you be Surprised?
Weather hidden or not, the party is "unaware of a threat" and will be surprised. This lasts until each character's first turn is over - there is no "surprise round" in D&D 5e. Note that this may be before or after the Assassin Vine's turn.
There is nothing to suggest that Entangling Vines requires the Assassin Vine to move. If it doesn't move then it continues to have False Appearance and if already Hidden, it stays Hidden.