"What would you like to buy?"
I'm sure someone else could tell you more, but I can tell you that 想 and 要 have ranges of meanings, and that those ranges overlap.
想 can mean： "want" (or as they say here, "would like"), "miss (someone)," and * "think (in the sense of thinking about, mulling over)."
要 can mean： "want," "need" (the two syllable word 需要 specifically means need, but 要 alone can mean "need," which I think of as a sort of abbreviation of 需要. Furthermore, 不要 means "need not" or "don't have to.") * "will," indicating the future tense, often in the sense of a future event that that you will make happen through your action or desire (as opposed to something outside of you that will occur in a more neutral/objective way, in those cases you use 会). In a sense, using 要 this way is almost like describing something you want to have happen in the future, or something that is going to happen in the future because you want/decided/are working towards it. That's a slippery rule, though, because 要 is often used in the structure “ 快要...了” meaning something will happen soon, (whether it's something you are bringing about or not).
I believe 想要 is just a classic case of the structure of modern Chinese, which likes to take two monosyllabic words and put them together to build a two-syllable word that combines or clarifies their meanings. I think 想要 just clearly means "want" (as in, it does not mean "thinking about," or "will in the future". It sounds a bit redundant though and in my experience most people will usually use just 想 or 要.
My personal, totally unofficial sense of it is that using 想 for "want" has more of a feeling of longing to it, and using 要 for "want" has more of a this-is-going-to-happen feeling to it. So my advice is, be aware of the range of meanings, and pick what fits best or avoids confusion.
Everybody, please feel free to add to or correct my definitions!
Gabrielle145359 explained it very well, i just wanted to clarify that 不要 means "don't" as in 不要走！(don't go!), but 不需要 or also 不用 mean "don't need".
要 can be a bit tricky since depending on the context it can mean different things, eg.: 我今天要上班 - I have to work today. 我要一只猫 - I want a cat. 我要休息一会 - I want to rest a bit. 我要去超市 - I want/need/am going to go the supermarket.
想 can also mean "to think" (你在想什么？ What are you thinking about?) or "to miss someone" (我很想你 - I miss you) in addition to "would like to".
When talking about what you want to do, you can use 想 or 要, when talking about which things you want you can use 想要 or 要. Just be careful with 想要 because in certain situations it can have sexual connotations (so it's ok to 想 your friends, but probably not to 想要 them).
But in this particular sentence, you could use both 想 and 要 without changing the meaning very much.