What's the differences between won't and will not? Do they have the exact same meaning in all contexts?
If not, I would really appreciate some examples.
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Won't is simply a contraction of the words will not. They have the exact same meaning. Won't is more informal; if you're writing an essay, in most cases you're advised not to use any contractions. Beyond that, there's no reason not to choose whichever you like. More often when speaking, you'll hear won't. So if you're writing dialogue, you might use the contraction to make it sound more natural.
Contractions such as ‘won’t’ are found principally in speech and in informal writing, although there seems to be a growing trend for them to occur in formal writing as well.
Where the full form does occur in speech, it is often used for exaggerated emphasis. ‘I WILL NOT GO’ spoken slowly and deliberately shows greater determination than ‘I won’t go’ spoken softly.
"Won't" is a contraction for "will not", and so they should mean the same thing in all cases. "Will not" sounds more formal and emphatic.
There is one case where it's not exactly the same, and that is when you're asking a question. The word order changes, and "won't" can contract a "will" and "not" which are separate:
Casual: "Won't you help me with this?"
(Very) Formal: "Will you not help me with this?"
"Will not you help me with this?"
In all cases a contraction sounds less formal. The strange thing about "won't" is that it contracts "will not" and so why isn't it "willn't"? the word for "will" used to be "woll" and the contraction evolved into "won't" before "will" supplanted "woll" in the vernacular. There is an intermediate version of the contraction, "wo'n't", which illustrates what we expect from a contraction... In this case the missing "ll" replaced by an apostrophe as well as the missing "o".
The question may arise why "will not" is contracted to "won't" and not to "willn't/ wiln't/win't". The /i/ in "will" is a secondary sound, the primary sound was o as we see in Latin vole:re (infinitive, to will/modal verb will). The present tense of volere has vowel change due to endings with i.
Latin vol-o - I will
Latin vis - you will (vis contracted from *vol-is. The form was used so frequently that it was rapidly shortened.)
We see this vowel change also in German: wollen - wir/sie wollen - ihr wollt - infinitive - we/they will - you will, plural but
ich will - du willst - I will - you will, singular.
I didn't check how things were in Old English. It would be interesting to see whether there are forms with o such as woll to be found and when secondary forms with i occurred.
The o is also in volition.