"Neither" is the same as "not either." If you have two options, and you want to negate both of them, you use "neither."
"I like Jenny and Ruth, but neither one likes me." (I like Jenny. I like Ruth. Both Jenny and Ruth do not like me.)
"I invited John and Bobby to the party, but neither of them came." (I invited John. I invited Bobby. Both John and Bobby did not come to the party.)
"I ate the pizza and pie, but neither one tasted good." (I ate the pizza. I ate the pie. Both the pizza and the pie did not taste good.)
"Neither one of us knows the answer." (I don't know the answer and you don't know the answer.)
"Too" is the same as "also," "as well," or "in addition." It's usually meant to affirm multiple positive statements. It can be used in various ways.
"I like Jenny, and I like Ruth too." (I like both Jenny and Ruth.)
A: "I scored 100 on my test yesterday."
B: "I did too!" (We both scored 100.)
A: "I have ketchup on my hot dog."
B: "I have mustard too." (Like you, I have ketchup, but I also have mustard.)
Do these examples help you understand?