The problem I see with the sorcerers having so few spells known is that it quickly becomes a matter of making the right choice. You have no room to make sub-optimal choices that might be more in theme with your sorcerer.
It can look that way, but OTOH, if you /can/ resist that temptation at chargen and level-up, you're good. From then on, you make the optimal choice of which spell known to cast each round of each encounter of each day until you level again (then you maybe succumb to temptation and retrain one flavorful spell for one optimal one).
The wizard is much tastier system-mastery bait, and every day it comes back and whispers "memorize -er prepare an optimal slate of spells!" and whenever an opportunity to learn a new spell materializes, there it is again "go ahead, learn the spell, it's a good one, who cares what your 'concept' is? You're a wizard!"
OTOH, it's less pronounced than it was in 3.x, when spontaneous casting was the Sorcerer's thing, not also the wizards, and you got more slots/day to play with.
This forces system mastery, which is supposed to be one of the things 5e was made to address.
It does, though, as I said, maybe less than it seems to, and system mastery is always going to be a thing, and 5e does address it with BA and by the simple expedient of having far fewer player-facing choices than 3e (not just in the obvious ways, by not having tons of feats, or PrCs or whatever, but by eschewing wealth/level, putting magic items very much back in the DM's bailiwick, and generally putting all resolution in the DM's hands).
Another potentially interesting way to think about this is:
Sorcerers are like the X-Men. Each Sorcerer (X-Man) has his own unique shtick, different from other Sorcerers (X-Men). The Sorcerer spell list is not the spell list of a single specific sorcerer, but rather an entire menu of what is possible to accomplish using sorcery.
That was so true in 3.x - I mean, literally true: there were Sorcerer builds of the X-men floating around gleemax. IMHO, precisely for the reasons I mentioned to Hawk, above.
It's less true in 5e because of the loss of extra slots and everyone being spontaneous - and because of sub-classes hitting the vague-3.5-suggestion of draconic heritage and the 4e chaos-sorcerer 'build' (really just a reprise of the 2e Wild Mage) so hard - if justifiably so in the name of concept-first flavorful design.
Wizards, on the other hand, are... well, wizards. Like Mazirian the Magician
Heh, just like him, and unlike any other 'wizard' outside of Vance & D&D fic, yeah.

(Well, not just like Mizirian - among the great magicians of his age and only able to memorize half a dozen spells - like Gandalf, he must have been around 5th level! "Did I say Prismatic Spray? I must've meant Chromatic...")
(OK, now I have to play a Wizard named Wizirian, because: alliteration.
Yeah, who only preps his half-dozen highest-level spells & never uses cantrips, because that's all just beneath his dignity.)
they strive to "collect them all". The Wizard spell list could theoretically be the spell list of a single specific archmage (like Mordenkainen or Phaandaal), even though a given wizard will usually only be a small fraction of the way along that path. In this sense, wizards are kind of interchangeable and undifferentiated from each other: a 20th level wizard is much more similar to another 20th level wizard than a 20th level Sorcerer is to another 20th level Sorcerer.
Yes, very true. And, by the same token, the Wizard is a stronger case (than the Sorcerer) of 'forcing system mastery,' because the opportunity to optimize is there every time you prepare spells and every time you have an opportunity to learn a new one.
(1) The Sorcerer spell list is overly constrained, and pretty uninteresting compared to the wizard list.
Nod. In 3.x it was a sub-set of the Wizard list, but only by 2 spells that would be meaningless if you didn't prep. Like the Bard, it'd make sense for Sorcerers to be able to poach spells from other classes - if they fit their Heritage.
The PHB is very generous with wizard spells. You automatically learn two spells each time you go up a level, so in practice 5E wizards don't have to behave like Mazirian and covet/swap/trade/steal spells from other wizards, since they already have their most-coveted spells of each level automatically.
OK, sure. Especially compared to the olden days when you started out with Read Magic, three random 1st level spells, and that was it, you had to find anything above that (maybe you got one new spell when you gained a new spell level?) or engage in insanely expensive and time-consuming 'spell research' (which, if you ever did, you might as well try for a completely unique spell!).
You could target either or both of these pain points if you want to more fully realize the differentiation between them.
I think the 3e differentiation worked: Make the Wizard prep into his slots instead of cast spontaneously. Give the Sorcerer a few more slots.
For Sorcerers, I have no particular problem with just letting a given sorcerer pick any effect which could be researched fairly easily by a wizard, and make that his sorcerer spell.
Assuming the new spell fits his heritage, sure. Nice idea.