If a hidden creature fails to surprise a character, is the creature still hidden?
For example, a goblin is hiding in some bushes, and a PC walks by. To determine surprise, the DM rolls for the goblin's stealth check, and the result is 8. (The d20 roll was a 2, plus the Stealth modifier of +6.) The PC's passive perception is 16, so the goblin doesn't surprise the PC, but is the Goblin still hidden?
In your example, the goblin is trying to hide from the PC. If its Dexterity (Stealth) check is lower than the PC's passive Perception, then its attempt to hide fails and its presence is noticed. It follows the same rules as any attempt at hiding; if another creature's Wisdom (Perception) check or Passive Perception beats your Dexterity (Stealth) check, then they notice your presence.
For reference, see the "Hiding" sidebar below this section of the basic rules:
The DM decides when circumstances are appropriate for hiding. When you try to hide, make a Dexterity (Stealth) check. Until you are discovered or you stop hiding, that check's total is contested by the Wisdom (Perception) check of any creature that actively searches for signs of your presence.
You can't hide from a creature that can see you clearly, and you give away your position if you make noise, such as shouting a warning or knocking over a vase. An invisible creature can always try to hide. Signs of its passage might still be noticed, and it does have to stay quiet.
In combat, most creatures stay alert for signs of danger all around, so if you come out of hiding and approach a creature, it usually sees you. However, under certain circumstances, the DM might allow you to stay hidden as you approach a creature that is distracted, allowing you to gain advantage on an attack roll before you are seen.
Passive Perception. When you hide, there's a chance someone will notice you even if they aren't searching. To determine whether such a creature notices you, the DM compares your Dexterity (Stealth) check with that creature's passive Wisdom (Perception) score, which equals 10 + the creature's Wisdom modifier, as well as any other bonuses or penalties. If the creature has advantage, add 5. For disadvantage, subtract 5. For example, if a 1st-level character (with a proficiency bonus of +2) has a Wisdom of 15 (a +2 modifier) and proficiency in Perception, he or she has a passive Wisdom (Perception) of 14.
Basically, hiding is generally a subset of surprise; hiding might enable you to surprise someone. If they do detect you, then by definition, you're not hidden.
The piece you're missing is that being hidden is what would permit the goblin to surprise the PC in the first place. By the same logic, being detected (in your example, by failing to beat the PC's Passive Perception) is precisely what causes the goblin to lose the benefit of surprise. If the goblin had remained hidden, it would have surprised the PC.
See PHB p. 189, which states:
Any character or monster that doesn't notice a threat is surprised at the start of the encounter.
(Emphasis mine.) The act of noticing happens when the goblin's Stealth fails to beat the PC's Perception. It is hard to imagine why or how a PC could notice a threat for purposes of surprise, but not for purposes of hiding.
Surprise, per the Player's Handbook:
Any character or monster that doesn't notice a threat is surprised at the start of the encounter. If you're surprised, you can't move or take an action on your first turn of the combat, and you can't take a reaction until that turn ends.
Surprise itself isn't necessarily the roll you're making. The roll you're using to is whether or not something notices you (aka stealth). But you can just as well surprise someone by pretending to seduce them and then ganking them with a knife when you get close (which would use Performance as the main roll).
In this case, you are attempting to surprise the enemy with Stealth as your condition for the Surprise. If your roll fails, your stealth fails, and so the jig is up, and you're now spotted.
