The last time I played a D&D game with actual humans around an actual table, Sorcerers didn't exist and Bards were a "prestige" class (meaning you had to jump through hoops to play one). Thus, I never developed an intuition for either class. To me, both 5e classes seem to be "discount wizards with benefits," which suggests I'm missing something.
I got it in my head that I should try a Sorcerer playthrough (specifically Storm Sorcerer), but 4 levels in, I"m tempted to just re-roll Wizard, mainly for the wider spell selections (both in class spells and in the Wizard's ability to learn spells outside the class). I suspect I'd have a similar reaction to Bards, but I'm not yet tempted to play one yet (though, if I'm reading things correctly, it seems that Bards, at least, have some unique spells).
Anyway, before I get into old man ramble mode, I wouldn't mind getting different viewpoints on what 5e Sorcerers and Bards really are, and, specifically, why they're not just the "budget Wizards" that my brain insists they are.