이에요 vs. 인데요
이에요 is a plain sentence ending with no special nuance. It just tells something in a matter-of-fact way.
(-입니다, -이다, -이에요, and -이야 are all similar except for the different levels of politeness and formality)
인데요 is 이다 ("is" - the coupler) connected with the conjunction-like verb ending -ㄴ데(요), which can range from "when" to "but" in meaning. That is, -ㄴ데 has the function of putting the verb in opposing relationship with the clause that follows it, similar to "considering this, how come?" or "given this, that is rather ...", etc. So if you end a sentnece with -ㄴ데(요) without the resolving second clause, it sounds like the sentence is unfinished, as if asking or challenging the listner to respond.
- 오늘은 학교 일찍 가는 날이에요 = Today I am supposed to go to school early. (states the fact plainly)
- 엄마: 자 빨리 일어나. 학교 갈 시간이다 = Wake up now. It's time to go to school.
- 딸: 오늘 토요일인데요? (= 오늘 토요일인데 (왜 학교 가래요)?) = It's Saturday (so why are you telling me to go to school)?