I am in a campaign with two Sorcerers. The players like the class... There is basically the Fire Sorcerer (Red Draconic Origin) and the Lightening Sorcerer (Tempest Origin). They use about 3-5 spells each - e.g., Fireball, Firebolt, Fly for the Fire Sorcerer. The limited spell list forces the character to focus and the origin provides some more interesting capabilities.
IMO, Sorcerer is for magic users that want to focus. Wizard can be for magic users that want to be spell monkeys - the magical equivalent of skill monkeys.
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.
Wizards, on the other hand, are... well, wizards. Like Mazirian the Magician, 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.
Or at least, that's the way it SHOULD be. What gums this up is twofold:
(1) The Sorcerer spell list is overly constrained, and pretty uninteresting compared to the wizard list. Much of the differentiation between spells is stupid stuff like different ways to inflict 10d8 damage. A DM has got a lot of work cut out for him if he's trying to provide multiple ways for a Sorcerer to be awesome. Third-party products like the Book of Lost Spells can help.
(2) 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.
You could target either or both of these pain points if you want to more fully realize the differentiation between them.
Worthwhile house rule: wizards get no spells automatically when they level up, but they DO start with a book of spells as starting equipment (with say, 1st through 5th level spells in it), and furthermore the DM invents spell research rules so that wizards are guaranteed to be able to convert gold and time into new spells if they like. (There may not be magic item shops, but there ARE spell research libraries and maybe even universities!) Those new spells do not have to be identical to PHB spells. If you want to research Acidball instead of Fireball or Cone of Heat instead of Cone of Cold, be my guest. Here's the DC and the number of weeks it will take. If you want Dimension Door to work on unwilling targets, be my guest. Here's the DC and number of weeks. Etc.
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. You want a no-material-components version of Stoneskin that turns you to steel instead of rock and lasts only for one minute while increasing your Strength by +2 at the same time? No problem, that's a fifth level Sorcerer spell, you can pick it at 9th level. Etc.