Wikipedia summarizes the distinction between formal and informal fallacies,
A formal fallacy can be expressed neatly in a standard system of logic, such as propositional logic, while an informal fallacy originates in an error in reasoning other than an improper logical form. Arguments containing informal fallacies may be formally valid, but still fallacious.Wikipedia contributors. (2020, October 9). Fallacy. In Wikipedia, The Free Encyclopedia. Retrieved 03:07, October 14, 2020, from https://en.wikipedia.org/w/index.php?title=Fallacy&oldid=982577448
To understand formal fallacies, we must first understand examples of formal logic. Good examples of formal logic include modus ponens or modus tollens. Modus ponens and modus tollens involve conditional statements like if and then. So the if part contains the antecedent and the then part contains the consequent. So, “It is Monday” is the antecedent and “It rains” is the consequent in the following conditional, “If it is Monday, then it rains.” An example of using modus ponens would be “If it is Monday, then it rains. It is Monday. Therefore, it rains.” An example of modus tollens would be “If it is Monday, then it rains. It is not raining, therefore it’s not Monday.” Occasionally people get formal logic wrong. An example is affirming the consequent. Given, “if it’s Monday, then it rains,” it does not follow to say, “it rains, therefore it’s Monday” since it could also rain on Tuesday. Another example is denying the antecedent. Given, “if it’s Monday, then it rains,” it does not follow to say, “it’s not Monday, therefore it doesn’t rain” since it could still rain on Tuesday or Wednesday.
The bizarre thing is that a formal argument can be valid but informally fallacious. So say you’re given two choices, A or B and you can only pick one but not the other, it’s usually referred to an exclusive OR. So if you choose A, then you can’t have B. So A, therefore not B. Also, not B, therefore A. It’s totally formally valid. So if a candidate said, “You either don’t care about the future or you would vote for me for President.” That’s actually formally valid. Basically it’s setting up the conditional, if you care about the future (which most people do), then you’ll vote for candidate A for President. The only problem is is that it’s a false dilemma since you’re not actually forced into just those two choices. It’s a false dilemma because it omits other possible options. Since it’s a false dilemma, then it is informally fallacious even though it is formally valid. Sure, if you only have two options, then in terms of formal logic, you know if one is false the other must be true in A or B situation.
People are actually pretty good at formal logic. They use it all the time and the only time they screw it up is when they’re in a logic class since sometimes you have to look at some complex sentences. Most people know that if you can only have either A or B, if you choose one you can’t have the other. But most of the time people commit informal fallacies, which usually means confusion over meaning, being lazy and using circular reasoning, thinking that there’s 2 or 3 choices when there’s more. To avoid informal fallacies requires a lot of attention to detail since some terms are loaded like “communism” and some terms are more straightforward like “dentistry.”