Working through IM Jeremy Silman's The Amateur's Mind, and the first proper chapter is on the battle between knights and bishops. Five rules in, and there's already a theme, a piece's value isn't fixed, it depends entirely on what's happening around it. Here's what I took from each one.
Rule 1: Piece value depends on position, not on the label
Bishops and knights are both worth roughly three points on paper, but you can manipulate a position to make either one far more or far less valuable than that. Silman's first example shows a bad bishop stuck behind its own pawn chain, completely unable to affect the game, while a knight in the same position turns out to be invaluable, hopping around to help push a pawn through to promotion.
Rule 2: Bishops are best in open positions, unblocked by pawns
Using a position from an accepted Danish Gambit, Silman shows just how dominant a pair of active bishops can be once the pieces are developed and the position is open, even while three pawns down. The engine apparently only rates the position as about a pawn worse overall, because the two bishops are doing that much work.
Rule 3: Bishops beat knights in endgames with advanced passed pawns
Silman's example here has White with connected passed pawns running down the board, but Black's single bishop is enough to stop the lot of them advancing any further. Meanwhile Black's own passed pawn is unstoppable, because White's knight is stuck doing nothing in the corner.
Rule 4: A bad bishop can still be an active bishop
A "bad bishop" is one stuck on the same colour squares as your own central pawns, limiting its activity. Silman's suggested fixes are to trade it off, move the pawns, or get the bishop outside the pawn chain entirely. His example shows a bishop that's technically bad by definition, same colour as its central pawns, but still active because it sits outside the pawn chain. He calls this a "bad active bishop."
Rule 5: A bishop's biggest weakness is that it only sees one colour
Silman's final example in this section shows two passed pawns for White being fully blockaded by Black's bishop and king together, and because the king sits on the opposite colour square to White's bishop, White has no way to chase it off.
Where this leaves me
Five rules in and the throughline is obvious in hindsight, neither piece is just "worth three," it's entirely about what the position is asking of it. I've definitely played games where I've clung onto a bishop or a knight out of habit rather than actually asking which one the position wanted. Something to start checking for. Part 2 to follow once I'm further into the chapter.