What is the difference between 该 and 应该
I first learned the phrase 应该 when talking about things that “should" be done. For example:

今天我应该学习中文。

But, when searching for some exercises to practice my Chinese, I came across the phrase:

我该怎么称呼你？

Is it also possible to say 我应该怎么称呼你？or does this sound strange at all?

I'd appreciate any help in understanding this. Thank you. :D

Question

What is the difference between 该 and 应该
I first learned the phrase 应该 when talking about things that “should" be done. For example:

今天我应该学习中文。

But, when searching for some exercises to practice my Chinese, I came across the phrase:

我该怎么称呼你？

Is it also possible to say 我应该怎么称呼你？or does this sound strange at all?

I'd appreciate any help in understanding this. Thank you. :D

Answer

I disagree with Stephen. 
該 is simple a short form of 應該, and is often used in speech. 
我應該怎麼稱呼你 is absolutely fine.

Answer

应该 and 该 is the same meanig  of the word, could be "should" in this sentence . So both can say.
应该 is more formal way and 该 is more casual way to express , but 该 sounds much stronger and direct than 应该 .

Answer

該,also means owe : 
我該你一元錢 i owe you 1$
with "應"to limit, "應該" strictly meaning "should"
similar to this, "在" and "正在"

Answer

就像是need 和should 一样
Need 就像 应该   语气委婉 
Should seems to be 该  语气强硬

Wish that could help you