Both of these words can have different meanings depending on the context. I can give you a better explanation if you tell me the whole sentence. Зажлалгүй has the root verb зажил (зажлах) and the suffix -лгүй. The official name of this suffix is "үгүйсгэн холбох," which means negative connection. So this suffix usually connects the verb to the next clause while negating the verb itself. Sometimes this suffix can be translated into English as "without." For example, "хоолоо зажлалгүй идэх" - to eat the food without chewing. Зажлахгүй just means "will not chew" (it can also imply present tense depending on the context). Hope it helps!