'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.

Neither = not + either. It's that simple.

"I enjoy either football or tennis" and "I don't want to play neaither tennis or football"

Both are used interchangeably in spoken English in certain contexts. To me though it sounds clumsy. 'Either' is used: When we have a choice. You have a choice - you can have either the salmon or the porc. It means one or the other. When we qualify an extra thought about something. - She is an idiot. And she smells. And i don't like her either. It means 'as well'. 'Neither' is used when you have a choice and don't want either one. You want the salmon or the porc? Neither. When saying two things don't do something. Neither of them work. When saying you don't want something. I hate them, and i won't be going on holiday with them neither. This is a case where 'either' is probably better but in spoken English you'll hear neither plenty. One last thing is that neither and either are pronounced two separate ways. You can say 'nigh-thur' or 'nee-thur'. I use both interchangeably depending on which one randomly comes out of my mouth. I don't know i'm doing it either.