Knowing, Preparing, and Casting are not the same thing...
While the spell slots progression is same1 for each type of Spellcaster2 (full, half, third), what spells they know and prepare vary by class.
- Druids, Clerics, and Paladins know every spell on their class list, and prepare a subset of them each day. Some paths have a number of spells (potentially off-class) that are always prepared.
- Wizards know a subset of their class list as contained in their spellbook; given time, the spellbook can include the entire set of wizard spells. They prepare a limited number of those spells each day.
- Sorcerers, Bards, Arcane Tricksters, and Eldritch Knights know a subset of their class list and have all of them prepared at all times.3
All spellcasters cast spells in the same way: select a spell from their list of prepared spells, select an available spell slot of the appropriate level, and then cast the spell.
Spells remain prepared until the next long rest, when the character can shuffle spells around - casting doesn't affect the list of prepared spells. A character with four spells prepared and two slots could cast the same spell twice, or two different spells (once each).
1Disregarding multiclassing, for simplicity's sake.
2Despite similarities, a Warlock's Pact Magic is not Spellcasting. Again, I'm leaving it out for simplicity's sake.
3Technically, they don't prepare at all, but it's easier to explain if you assume they're all always "prepared".