The います in 違います comes from the way the verb 違う is conjugated with the polite ending ます. To put a verb that ends in う in the ます form, the う is changed to い.
This is unrelated to the verb of existence いる (います in polite form). And since it's a verb conjugation, it's not possible to replace the います of 違います with です.
Here are some other common verbs that end in う and their ます form.
- 会う > 会います
- 言う > 言います
- 歌う > 歌います
- 買う > 買います
- 使う > 使います
The English translations of "that's incorrect" (using an adjective) or "there is a difference" (a noun plus the copula) offer a hint to why you might have overlooked the possibility that this was a verb conjugation. A more literal translation that uses an equivalent verb in English might be "it differs". Sometimes translations do not maintain the same part of speech as the original text, in order to remain natural for the situation in question.