Would someone please clarify the importance of the distinction between
a) either A is true, or Not A is true
b) either A is true, or A is not true
I've seen answers on this site in which the difference between those two formulations is very important, but I'm not quite sure why.
It would be helpful to know which answers on the site you're referring to, but I'll take a stab at your question anyway. The only difference between your formulations (a) and (b) is the second disjunct in each, so I'll focus on that. I presume A is some statement. What's the difference between "Not A is true" and "A is not true"? I'm not sure there's always a difference. Depending on the system of logic or semantics, "Not A is true" can mean merely (i) "Whatever truth-value (if any) A has, it's not the value true ." Or it can mean (ii) "A is false " in systems in which every statement is true or false. The difference between (i) and (ii) is sometimes important, such as when we're dealing with the classic Liar sentence (L) "This sentence is false." One might say that L is not true and yet not false either: one might say that L is neither true nor false.