5e doesn't specify how to simulate this, so it's up to the DM
5th edition assumes that Deafened condition is temporary (hence the name) and, unlike the previous versions, have little mechanical downsides for it. There is no failure chance when you cast a spell being deafened in 5e. There was one in 3.5e/Pathfinder though:
a spellcaster who has been deafened has a 20% chance of spoiling any spell with a verbal component that he tries to cast
Possible RAW mechanical downsides of being deafened in combat were already discussed in this question: How does deafness affect melee combat?
There is no rules-as-written penalty to Perception check either, since 5e almost got rid of any check penalties, using disadvantage mechanics instead. You just can or can not perceive something, depends on the sense used (sight or hearing). The final verdict is always up to the DM. The DM might skip the check, or impose a disadvantage if they thinks it is necessary.
The rules in 5e don't explicitly codify many things. The DM is encouraged to use common sense instead and apply disadvantage when it is appropriate. For instance, if your group doesn't use the Flanking optional rule, it can be used for the deaf person exclusively, since he have troubles hearing enemies creeping from behind.
There are almost no advantages of being permanently deaf
The implications of being deaf are pretty obvious, although it's hardly possible to make "the complete list". A deaf person can't hear anything. Many "negative effects" would be narrative rather than mechanical ones (this can be applied to 5e in general). For instance, you can't hear people talking - so no eavesdropping, no bargaining nor gathering information in taverns, no parleying with goblins, etc.
Having the sign language known only by the party isn't the positive side of being deaf. It's a positive side of knowing such a language — you don't have to be deaf to know it.
The only positive side of deafness is the immunity to any spells requiring the target to hear. For example, the Dissonant Whispers spell explicitly says it doesn't work on deafened targets:
A deafened creature automatically succeeds on the save.
There are other spells, requiring "a creature that can hear", like Suggestion:
You suggest a course of activity (limited to a sentence or two) and magically influence a creature you can see within range that can hear and understand you.
PH has a few spells with this requirement:
- Animal Friendship
- Dissonant Whispers
- Divine Word
- Suggestion/Mass Suggestion
- Tongues (this one actually has a positive effect)
- Vicious Mockery
Additional features using this requirement:
- Bardic Inspiration (also have a positive effect)