Comparison
yào
modal verb
xiǎng
modal verb

HSK1 TBCL2
expressing desire or willingness

The modal verbs xiǎng 想 and yào 要 both indicate desire or willingness, as if saying “would like to do something” or “want to do something”.

Differences

While being used in the same context, yào 要 sounds more determined and direct, whereas xiǎng 想 implies an intention or desire that may result from consideration, which sounds less direct and more polite.

我要喝酒。
我要喝酒。
Wǒ yào hē jiǔ.
I want to have some wine.
(Implication: I have a strong desire to drink)
我想喝酒。
我想喝酒。
Wǒ xiǎng hē jiǔ.
I would like to have some wine.
你要喝酒吗?
你要喝酒嗎?
Nǐ yào hē jiǔ ma? 
Do you want to have some wine?
你想喝酒吗?
你想喝酒嗎?
Nǐ xiǎng hē jiǔ ma? 
Would you like to have some wine?

Compared to yào 要,  xiǎng 想 is more compatible with the tag question kěyǐ ma 可以吗 ‘can I, may I’, to form a polite request.

我想喝酒,可以吗?
我想喝酒,可以嗎?
Wǒ xiǎng hē jiǔ, kěyǐ ma?
I want to drink liquor/ I would like to drink liquor, may I?

Compared to bú yào 不要, the negation form bù xiǎng 不想 is more appropriate for a refusal to make it less direct and more polite.

A: 你想吃这个吗?
A: 你想吃這個嗎?
Nǐ xiǎng chī zhè ge ma?
Would you like to eat this?
B: 谢谢,我不想吃。
B: 謝謝,我不想吃。
Xièxie, wǒ bù xiǎng chī.
Thanks. I don’t want to have it.

In this use, xiǎng 想 can be modified by degree adverbs or degree adverbial phrases such as hěn 很 ‘very’, zhēn 真 ‘really’ or bú tài 不太 ‘not very’, but yào 要 cannot.

姐姐现在很想休息。
姐姐現在很想休息。
Jiějie xiànzài hěn xiǎng xiūxi.
My elder sister really wants to take a rest now.
我的朋友不太想去打篮球。
我的朋友不太想去打籃球。
Wǒ de péngyou bú tài xiǎng qù dǎ lánqiú.
My friends don’t really want to go playing basketball.