있어요 and 이에요 are the present tense, polite speech conjugations of their root verbs 있다 and 이다. - 있다 => 있습니다(formal-polite) / 있다(formal-plain) / *있어요(informal-polite) / 있어(informal-plain). - 이다 => 입니다(f-polite) / 이다(f-plain) / *이에요(i-polite) / 이야(i-plain). (formality is about how rigid or smooth it sounds, and politeness about showing respect to the listener) 있다 and 이다 are two different verbs that together correspond to the function of "be" in English. 1. 있다: like "is" in "There is AA" (AA가 있다). It means AA exists. 2. 이다: like "is" in "AA is BB" (AA는 BB이다). It equates AA with BB. 있다 is like any other verb in its usage. - 책상 위에 책이 있다: There is a book on the desk. (== 책상 위에 책이 있어요) - 문제는 우리의 내부에 있다: The problem is/lies within us. 이다 is a special verb (technically a "particle") which directly attaches to a noun with no space. - 나는 학생이다: I am a student. (== 나는 학생이에요) - 지는 것이 이기는 것이다: To lose is to win.

I don't really speak Korean well, but from my studies I can tell you that the first one is past tense of "to be" (was) and the second is present tense (is).