*Usually 别 is used to form an imperative sentence when you want to ask somebody not to do sth. E.g. 别吃=Don't eat! 别看电视=Don't watch TV！ *不 is often used to indicate a negative sentence. E.g. 我不走=I won't leave. 他不在这里=He's not here. However, 不要 is equaled to 别 when it indicates an imperative sentence. E.g. 不要吃=别吃，不要看电视=别看电视。 But 不要 can also means "not want or need sth" if the speaker doesn't mean to order others E.g. 我不要吃鱼= I don't want to eat fish. 他不要你帮忙= He needn't your help.

they both mean "don not eat" 不吃 is used by a person self,such as: i don not eat it. /he do not eat it.我不吃/他不吃。 别吃 is used for other person which you say this word to,for example:you don't eat it.你别吃。