There are certain instances where you might NEED to write a follow up letter like if the job is for the position of "Letter Writer" then maybe that would help out, or if you had to leave the interview early because you had killer diarrhea, then a follow up letter to explain you were taken to the local ER and wish to reschedule would be required. Aside from all that, I can't really bring myself to ever write a follow up response. What pisses me off about the whole thing is that it shouldn't even be a THING.
There should be no such thing as a "follow up letter", but thanks to one insecure, douchebag idiot somewhere in the past setting this precedent now EVERYONE is pressured to follow up. Thanks to this precedent and existence of a follow up, now it's almost a requirement after ever job interview and because everyone's doing it, then being the one person who doesn't might actually hurt your chances at landing the job. All of this because of one dude who couldn't just leave well enough alone, wanted to follow up and then got the job and told everyone and got that practice trending.
He could have just got the job and shut up about it, but instead he told his friends and his cousins and they told their friends and cousins until sending follow ups became this massive practice that's expected from everyone. It forces people to do unnecessary work, wasting millions and millions of dollars in resources and time for everyone! If I knew who that guy was I swear to god I'd beat him within an inch of his life with a bowling pin and then a week later send a follow up note to his hospital bed.