Learning by instruction is like following along while somebody connects the dots for you. Learning by discovery is like connecting the dots yourself.
Learning by discovery requires context, and builds context for further discovery at the same time. It grows, and grows out of, a solid foundation of understanding.
Instruction can lead you to new and unfamiliar places, but there is no guarantee that you will be able to find your way back to them.
Instruction typically requires 50 to 100 exposures in order to reliably stick. On the other hand, you only have to discover something 3 to 5 times to remember it for the rest of your life. Take, as an example, the beloved quadratic formula that so many of us memorize in school:
This formula is typically taught by instruction, so that you can get to work putting it to use. Various mnemonic devices and memorization schemes are employed. You might even remember it for a long time. But can you derive it? If you forgot it, could you deduce it again from basic principles? If somebody asked you what it is for or why it works, could you explain it? Could you teach them to do the same?
When you learn by discovery, you not only remember what you have learned, you also gain the ability to re-teach it.
Another difference between discovery and instruction is that instruction needs a teacher, but discovery doesn’t. Discovery is something you can do for yourself and by yourself.
The two are not mutually exclusive, however. In fact, the best kind of instruction is assisted discovery.