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Why does Esperanto use the letter Ŭ?
Hm. You didn’t ask why the letter looks like that, which I’ll answer anyway:
Wikipedia Ŭ suggests it was formed by analogy with proposed Byelorussian ў. Like someone else said on Wikipedia: 
Now, why <ŭ> and not just <u>? Zeibura, you dawg, you know that I love this kind of question, where I try to work out the answer from first principles. I’m pretty sure Gaston Waringhien has given a proper answer somewhere (after all, him and Kalocsay saw Zamenhof’s proto-Esperanto notes, before the Nazis torched everything). But let’s have some fun.
- In Zamenhof’s circumflexless (“telegraphic”) rendering of Esperanto, circumflexed letters get replaced by a following h: <ĉ> to <ch>. But <ŭ> could be rendered as just <u>. So Zamenhof was not overly concerned about ambiguity (and he normally was, which is why Esperanto is so neurotic about polysemy).
- I cannot come up with an minimal pair for au and aŭ or eu and eŭ. I’ve been racking my brains for an hour. For example, fra-ulo and fraŭlo would be different words; but there is no fra- root.
- Zamenhof had –aŭ as an odds-and-ends part of speech suffix; it dates from proto-Esperanto. The part of speech suffixes are otherwise vowels; so –aŭ in malgr-aŭ, bald-aŭ, apen-aŭ was deliberately intended to be a single syllable, just like bird-o, blank-a, kapt-i, plen-e.
- As presented in Duonvokaloj kaj diftongoj, Zamenhof’s advice to correspondents early on insisted on the phonetic difference between monosyllabic aŭ/eŭ and bisyllabic au/eu (which certainly does occur in Esperanto).
So, why <ŭ>? Theoretically it could have determined a minimal pair, but it certainly wouldn’t have with Zamenhof’s vocabulary, and I doubt it does even more. And Zamenhof wasn’t fussed about the ambiguity in his telegraphic rendering.
No, it was because in Zamenhof’s own mental model of the language, aŭ/eŭ were single syllables, just like aj/ej/oj/uj are. That’s why –aŭ is a part of speech ending. After all, <au> is a single syllable in German, and <αυ, ευ> in Ancient Greek—which is what Zamenhof must have had in mind in his design.
I think the question was why W isn’t used instead. That’s easy to answer by a short look at Polish, which like Esperanto has closed syllables ending in the consonant [j], but the rare diphthongs au and eu (rare enough to be somewhat shaky) do not end in [w].