“either” represents one of the two or any, “neither” indicates none of the two. Both are tangents in comparison.
‘Either’ is considered to be the positive pronoun, adverb, conjunction or adjective as per its usage. It provides the audience with the knowledge of expecting in advance two or more possibilities during the conversation or dialogue. On usage either is always paired with ‘or’. When there are more than two options then usage of ‘any’ is preferred in that context. As an adjective, ‘either’ can be used only between two options hence it is one or the other, or both of them. For example, as an adjective it is used as “Either side of the river”. As a pronoun, it is one or the other hence sample usage is “Both the cars are going home, you can drive in either of them.” As a conjunction it gives a possibility of choice, therefore it can be used as “Either email or watch TV.” Though it is positive in usage, as an adverb it follows a negative frame of words like “If you don’t play, the dog won’t play either.”
On the other hand, ‘neither’ is considered to be a negative pronoun, adjective, conjunction and adverb. Always paired with ‘nor’ it sets the platform for the audience to not consider any of the possibilities provided during the conversation. As a pronoun, ‘neither’ is used as “Neither of the pups are of a pure breed” while as an adjective it is adapted as “Neither flowers are fresh.” As a conjunction, ‘neither’ may be used as “Neither of the reports can be trusted.”