Между lots of и a lot of разликата е общо взето, че "a lot of" се смята за малко по-официално.

I have been to lots of festivals.
I have been to a lot of festivals.

==> значението е едно и също, все "много".

both - "и двамата", "и двете".

I have talked to both Jonas and Nick.
I have explained both theories to my students.

neither - нито/нито едно(то)/никой от.

Примери:
- Would you prefer juice or coffee?
- Neither, thank you.

- Did you two have a good time yesterday?
- Neither of us went to the party.

Ако искаш пък да кажеш "нито ....., нито ....." се комбинира с nor.
Neither Italy nor France got to the quarter finals last year.


Either e по същата "схема" , но с положителен смисъл.