For example, If you have two responses' Yes and Yes sir' for two people,

you will have to choose them to respond to them. 

Is it clear? 

which one is more polite? which one is more casual?

알겠습니다 is more polite(BUT, depends on your voice and tone.)

알았어요 is more casual(ALSO, depends on your face and attitude.) 

In Korea, there are many different methods... i think there are hundreds of thousands of different kinds of sentence types :) 

Anyway, I hope best of luck with your study

difference....? minimal

I'd say 알았어요 is more casual

알겠습니다 is more of.... very forrrrmal and strict...?

You'd say 알겠습니다 in workplace or in army ㅋㅋㅋㅋㅋㅋㅋㅋㅋㅋㅋㅋ

but whatever is fine

as I've said, the difference is miiiiniiiimalllll

두 문장의 차이는 분명합니다.

주종의 관계-주인과 종의-입니다.

당신이 주인인 경우 알았어요

당신이 종인 경우는 알겠습니다.

好运

"알겠습니다" is more formal and very polite. If you use "알았어요"to your teacher or  senior, you should show your positive facial expression or a respectful voice.

Both of them giving the same statement.