"That is not right, you know."
Whether or not it's looking for confirmation is sort of based on the tone in which it's said though. Sometimes it's used to express uncertainty, but it can also be used as a statement. Like, it would be the difference between "can you give me confirmation" and "take this opportunity to agree with what I'm saying".
When their English prompt is simply "you know", but よ and ね comes down to how you think "you know" is being used, so both should be accepted as the correct answer.
In proper Japanese sentence structure sentences end with 'desu', 'masu', etc. 'Masu' (or 'mashita' as the past tense) will always come after a verb. So while 'Chigaiyo' would be technically correct, it would only be so if you were on the streets talking with friends, while 'chigaimasuyo' would be the right way.
Actually, ちがいよ would technically be incorrect, as よ follows a rentaikei (連体形, or 'dictionary form'), which is ちがう (in plain) or ちがいます (in polite speech).
ちがい is called the renyôkei (連用形, 'using form'), and it's the form you use for most conjugations (except negations and commands). For example, you can see it's also used as the stem to paste -ます onto.
ちがいよ sería técnicamente incorrecto, porque la partícula よ debe ir despues de un rentaikei (連体形, o 'forma de diccionario'), que sería ちがう en informal o ちがいます en formal.
ちがい es un renyôkei (連用形, 'forma de uso'), y es la forma que se usa para la mayoría de las conjugaciones (excepto negativos e imperativos). Por ejemplo se puede ver que es una raiz a la que se le puede añadir el -ます.
Japanese people often say "ng" (the sound at the end of the English word "sing") for the sound that we write in romaji as "g".
There is an interesting discussion of it here:
What the ね does is make clear that the persons wants/needs confirmations or isn't sure about what they said. よ at the end of the sentence basically is a exclamation mark. Translating the よ as "you know" seems strang to me – but I also learnt Japanese to German, which makes a lot of the English translations here sound confusing to me.
~ます is a suffix that makes the phrase polite to the listener; it is added to what is officially called the 連用形, renyoukei, or also sometimes the i-stem (because for go-dan verbs (that have 5 different stems) it's the one ending in a vowel of the i-line).
So, the verb 違う, which is a go-dan verb, changes to 違い to have the sufix ~ます attached.
違います is a verb, in polite form, non-past tense.
です on the other hand, has two usages; one is to state equivalences between two nouns, or to state that A is a characteristic of B : ( [Bは]Aです。)
The other usage, is just to add a sense of politeness to sentencesn ending with an i-adjective (that are a lot like verbs, in that they change with tense, negative, etc.)
So, while when you add ~ます you just add politness level, but the verb remains the same and it's a verbal phrase: 違う = 違います = "it differs",
with です you need a noun, and you state something about that noun. A way to make a verb a noun is to add のこと (colloquially shortened to の or even ん) to the dictionary form of the verb, with a meaning like "the thing of doing..., the action of doing"
違うのです = it's the action of being different, it's different.
You convey the same meaning at the end, but using a verb or a noun of an action. "it differs" vs "it's different".
The grammar used changes, you cannot just switch ~ます and です, they are linguistically very different beasts.
I typed in 「違うですよ。」 I think was was incorrect because of the 「う」 and because of the 「です」. I thought I've heard people say 「違う’」to me in the past, which may be informal (not sure). Also, I figured [です」 for "That is" would fit, but apparently not.
No, it wouldn't be. Firstly, "tadashii" is an adjective, not a noun, so that should be "tadashikunai". Secondly, Duo 'teaches' neutral/polite speech, not casual/informal, so (if it were a noun) "ja nai" would be "ja arimasen". Lastly, what we wrote in Japanese was literally "it is different", which is often used to point out something is contrary to what the speaker thinks/said.