"That is not right."
~ます is a prefix used with verbs, that adds politeness to the listener.
です is used with nouns, in a way similar to the English "A is B"
So, grammatically they are different.
違う【ちがう】 is a verb, meaning "to differ". in polite form : 違います
そう is a word of the こ・そ・あ・ど system (こ~ = near the speaker, そ~ = near the listener, あ~ = far from both, ど~ question word). You have probably already seen words like ここ、そちら、それ、その、そんな...
Here ~う in the ko-so-a-do system makes a descriptive for a mood, a feeling. そう = the feeling near the listener = your feeling, like you think/said. Hence the common phrases そうですか = is it so?; そうですね = it is so (I share my feelings with yours).
どう is the interrogative word in the same family, so "how"
Now, directly negating the feelings or your interlocutor would be very rude; that is why you can use そうですね to agree with the listener, it would be rude to disagree by saying そうではありません
Instead, 違います, "it differs, it's the other way around" is used.
The よ on the end is actually a particle. Here it would be used to add emphasis, or impart new information. So ちがいますよ means the same thing overall as ちがいます, just with a bit of nuance.
Here's a link that talks about the two major ending particles: http://www.punipunijapan.com/japanese-particles-yo-ne/
And そうです can mean "That's right", too. Probably literally closer to "That's so," if you want to be more specific.
Be careful with this one ! :) ちがい looks a lot like an i-adj, but isn't one. It is a noun.
On a somewhat more advanced note, it's also the stem of the verb ちがう. You can get the stem of a verb by looking at the ~ます form and dropping the ~ます. So, the ちがいます stem is ちがい, たべます is たべ, のみます is のみ. I don't know that this is immediately useful here on Duolingo, but you do see these stem forms pop up in some grammatical constructions.
It should be ちがう (adj). Because ちがい (n) means "difference". But still that would just translate "it is wrong". それ わ ただしい ません Is what translates "that is not right (correct)"
Actually, ちがう is a verb. It's just the casual form of ちがいます. You can sometimes use it to modify nouns, like how in English "to jump" is a verb, but in "the jumping frog" it modifies a noun.
In your second sentence, ただしい is an adj, so it should conjugate as ただしくありません. The whole sentence should read, それはただしくありません. So you may notice I've changed the final い to a く and had to add the verb あります to negate the adj. ~ません is how you make a verb negative (in polite form), but it's only for verbs.
Semantically, as someone else noted in this thread, ちがいます literally means "to be different". It's just also often used in a context of, "That's different (from reality)." So "not right" probably comes closer to the literal meaning than "wrong", but translating it you could probably either.