想 and 想要 are basically the same. You could translate them as "would like to". I think the bigger difference lies between 想 and 要. Here's the tricky part: 要 is always taught to Mandarin learners as "want" but they don't learn the other meaning which means "to be going to" and they end up saying sentences like "我要去“ meaning "I am going to go.."when they really want to express "I want to go to.."
Generally, 想 and 要 are both used to express "want"
我想吃米饭 and 我要吃米饭 they both mean "I want to eat white rice", in which case are both interchangeable. However, in these examples they are NOT the same:
我想去中国 - I would like to go to China.
我要去中国 - I am going to China
要 is used to show that an action is definitely going to happen.It is also more affirmative and guaranteed. Imagine summer is here, school's out and you're talking about your plans.
你打算去哪里?=Where do you plan on going?
我要去中国度假三个星期 = I'm going on vacation to China for 3 weeks.
想 is more the thought of doing something. Take the following scenarios:
我想知道为什么,能不能告诉我?= I would like to know why, can you tell me?
我儿子喉咙痛,拉肚子,又发烧。我想给他请三天病假。-My son has a sore throat, diarrhea and a fever.I would like to give him three sick days.
Now let's take out "想" and stick in 要 and see what happens:
我儿子喉咙痛,拉肚子,又发烧。我要给他请三天病假 -My son has a sore throat, diarrhea, and a fever. I'm going to give him three sick days.
要 is more assertive and in this case there is no question on my actions. In the first example, 想 doesn't necessarily imply that I am going to give my son three sick days, but that I would like to do so.In which case, it is more of a request. In the second example 要 indicates strong intention and definite action.
想 also has other meanings. Such as to "think" 我在想办法 "I'm thinking of a method", 我想她 "i miss her".