What is the broad overview of the differences between Wizard and Sorceror in D&D 5e?

What are the differences between Wizards and Sorcerers in D&D 5e, in broad overview terms?

Really, I’d like to know this from two perspectives. One I have a somewhat idea about from reading the PHB and other websites, one I don’t get at all.

  1. What are the differences in the two classes from a character building background in terms of story and roleplaying? This is the part I don’t have any clue about.

  2. What are the differences from a mechanical standpoint? I understand Sorcerers get spells a bit quicker and it looks like Wizards have to buy their spells but I’m really not too clear on this either.

I’ve already read a few articles and watched a few videos but still don’t know the answer. I especially don’t know the answer to the character building / roleplaying side as most of what I did find just talks about which one is more powerful. These are the resources I’ve looked at thus far (not including the Player’s Handbook which I’ve read over and over now):

  • Why did the sorcerer/wizard spell lists diverge in D&D 5e?
  • Which spells do Wizards get at lower levels than Sorcerers?
  • Sorcerer vs. Wizard: Clearing Up the Choice 5e
  • Sorcerer vs. Wizard: Anything You Can Do I Can Do Better?
  • Nerdarchy’s YouTube Channel: Wizards Vs Sorcerer| Dungeons and Dragons 5th Edition Classes