The main difference is that "neither" is used for making negative statements while "either" is used to describe a choice or add emphasis to something previously mentioned.
Take a look at the following sets of sentences:
A) I can't attend the party. Neither can he. // I can't attend the party. He can't either.
Both function as linking words to connect the second sentence to the first. Here, "either" means "too" or "as well". Take note that since "neither" already has a negative connotation, we do not write "Neither cannot he".
B) There are parking spaces on either side of the road. // Neither of the boys is guilty of vandalism.
In B, both words function as pronouns that come before a noun (i.e. side, boys). Here, "neither of the boys" means "not one boy or the other". "either" means "one side or the other". As you can see, "neither" has a negative connotation while "either" has a positive connotation of having a choice between two options.
C) Either May or Paul can play. // Neither May nor Paul can play.
Here, "either" is paired with "or" while "neither" is paired with "nor". They now function as conjunctions. "Either May or Paul can play" means "May can play BUT Paul cannot play." or "Paul can play BUT May cannot play." (there is a choice between two options) "Neither May nor Paul can play." means "May cannot play AND Paul cannot play." (both choices are out)