Threats, in my view, mean danger in an immediate sense. Problems can lead to threats but are not imminently dangerous in the same way.
Resources and rewards don't break down in quite the same way.
Resources can take a lot of forms (this lever causes water in the dungeon to rise 5 feet, the Baron can be convinced to aid the party if they're polite, an Alchemist has set up shop by the crossroads). But they're things that the PCs have to choose to use.
Rewards also take a lot of forms (magic items, gold, cute girl feeding apples to your horses, banquet in your honor) but they're not things PCs are likely to pass up. If you think of magic items and gold as resources, then there is some overlap, but I think players do feel rewarded when acquiring those things, even if they proceed to use them resourcefully.
PTRR is about making sure you write stuff that will see use. In My Experience, when the GM puts something before the players, these 4 things are things the GM needs. I did a big diagram a while ago to determine the flow of play in my D&D games; I found after the player-characters are created and the players have accepted the basic premise (we're playing a game, your characters are fantasy heroes, here's a basic context for where you are) the DM's job was to put interactive elements in front of players.
Bad design, whether it's railroading, DMPCs, or boring settings, all have one thing in common: They put inert elements in front of players (a.k.a. they're all a waste of time).
"Here's the prince of Roundheria, he's thinking of invading the peaceful land of Overtheria, but you can't convince him not to."
"Here's the dragon of the west mountain, but he's so powerful he'll kill you all instantly, so listen to his monologue before he flies off to what I've already decided he's doing, OK?"
"You're in a town. There's an inn. No, nothing interesting is happening."
"Elminster talks to you for half an hour, here's his 20-page explanation of why he's not going to help you. No, you can't convince him."