Actually, 니다 is politer than 요. 먹습니다 is more polite than 먹어요, therefore. And 마십니다 is more polite than 마셔요. 다 is plain form. It's used mostly in writings (journal, books, newspapers, etc) and the news. It's plain, like it's called. So its neutral tone makes it polite enough not really conversational. You would sound boring and awkward and lifeless and emotionless. So when you're speaking, use the second one. If you're doing a more formal (or more accurately, official or literary) writing, use the first one. It's more suited for bookish writing.

Honestly, both verbs differentiate between whether it is formal or not. In addition, 어요 goes to formal. 다.goes to informal. So you are able to use ~다. with your friend or someone who is close to you. Moreover Koreans use 어요 to someone who is older than them or isn't friend with you.