봅니다 means wider than 읽습니다.
The basic form of 봅니다 is 보다, which can be used in this kind of situations,
1. what are you doing? / 뭐 해?
Just watching TV / TV 보고 있어. (means to watch)
2. What are you going to do tomorrow? 내일 뭐 할 거야?
I'm going to hang out with my firends. 친구들 볼 거야. (means to hang out with)
3. What will you do after work? / 일 끝나고 뭐 할거야?
I will just read a book. / 그냥 책 볼 거야 (means to read)
4. Is there anything speical outside the window? What are you looking at?
창 밖에 뭐 특별한 거 있어? / 뭘 보고 있는 거야? (means to see)
Nothing, it's likely to rain / 아무 것도 아니야, 비 올 거 같아.
And so on...
While, 읽습니다. The basic form is 읽다.
1. Have you read the book? 그 책 읽었어? (to read)
Yes, I've already done! 응, 벌써 읽었지!
2. I feel so sad now, no one would understand me.. 난 지금 무척 슬퍼, 누구도 나를 이해할 수 없을 거야.
Well.. I think I could read your feelings. 글쎄.. 난 네 마음을 읽을 수 있을 거 같은데.. (means to Know or see through)
I've just made some sentences in a hurry to explain the difference between two verbs. As you see, it's for sure that 봅니다 is much more used in conversations. It's not easy for me to explain something in English, even if I knew well about something, so it wouldn't be perfect to understand, but hope to be helpful. Good luck!