Both is a conjunction that can be used to join two nouns, pronouns, nominal phrases or noun clauses together when performing as either or complement of a predicate in any given sentence.
Consider:
Both the man and the woman who is standing over there belong to the same group. (Means both of them are in the same group)
In other hand, neither is also a conjunction but it is being used for negative sentence.
Neither the man nor the woman who is standing over there belongs to the group. (Means none of them is in the group)
Noted error: Using both in negative sentence instead of neither is wrong.
Always remember that both must be changed into neither in a negative sentence.
More examples:
Don't say: Both of them have not gone for the screening exercises.
Say: Neither of them has gone for the screening exercises.
Don't say: Both Mark and Anderson do not like me.
Say: Neither Mark nor Anderson likes me.