'Either' has positive connotations but 'neither' has negative connotations. There are many ways to use them.
"Do you like apples or bananas?"..."either" - in this case it means the person likes both of the items.
"Do you like grapes or kiwi?"..."neither" - this means the person does not like grapes or kiwi!
There are many other ways to use them.
"Today I will either go to the shop or the barber" - today I will go to one of these pkaces.
"Yesterday, I went to neither the gym or a restaurant" - Yesterday I did not go to any of these places.