I know there is little to no difference if you have Mayhem on but the one thing I thought was different in the past was that there is more Badass and anointed type enemies on TVHM.
Is that true or is that covered by Mayhem mode too? Back playing again for the first time in awhile awhile playing a FL4K for the first time and wondering if I need to do TVHM for that.
I remember back when i started playing that normal capped items at level 30 (can’t say if enemies did too)
And TVHM dropped quest items at the same level as your own.
Kinda easy->normal scaling haha but this game has had problems with difficulty scaling from the get-go (same for weapon balance)
To my knowledge as soon as Mayhem Mode is enabled, it becomes virtually identical. The only real reason to do TVHM is to potentially get a few DLC items at higher level if you did the DLC before you were max level or when the max level was still lower.
TVHM = you can reset your playthrough, anointed items can appear in-world (only available in Earl’s vendor in normal mode), greater elemental weaknesses/resistances kick in, world always scales to your level… yeah what @HammerSickle said, it’s basically Mayhem 0. 
If you’re intending on playing in Mayhem levels, the only thing you’ll be missing by not going to TVHM is losing the ability to reset your playthrough. Which of course some players don’t mind as they have no intention of playing through the vanilla and/or DLC story campaigns again.
Edit: there’s another perk to having a character go into TVHM: get that character as far as Sanctuary and park them there so you can have access to your bank. Farm for your stuff in ‘normal’ mode, then quit and restart in TVHM, throw your stuff in the bank, and then restart again back in normal mode and you’re right back at your farming spot, eliminating the chore of having to go through the map again for some farming spots.
As drops? Huh I coulda swore they were in Earl’s vendor only. Though as always I could be wrong.