From what I understand, 'både' is a conjunction and is often used with 'och.' So "I speak both English and Swedish" is "jag talar både engelska och svenska." Whereas 'båda' is a stand-alone pronoun ("Jag tycker om båda").
Here is a great article with explanation: http://www.thelocal.se/blogs/swedishteacher/2016/01/09/bade-or-bada/
Anyone, feel free to correct me if I'm wrong :)
No, you're certainly absolutely correct, but in this case of sides you're really talking about people on either side of an argument, not about actual sides. We use har fel for opinions and statements, whether expressed by a person or e.g. a newspaper or a political party. If you were talking about e.g. the sides of a cube being the wrong size, it would indeed be är fel.