5the quiet upgrade is the swing
It reaches further and swings faster than the plain net, but the part that changes how you play is that it cuts through grass, vines and other soft tiles as it passes. Bait critters spawn out of tall grass when you break it, and with this you break the grass and catch what jumps out in a single motion. No swapping to a sword and back. Gathering bait stops being a chore.
underworld critters and the hellstone trick
This and the Lavaproof Bug Net are the only nets that will hold lava critters, and this is the only one you can get without mining hellstone first. That opens a strange shortcut: those critters are bait, that bait pulls Obsidian Crates, and those crates contain hellstone. A patient angler can be smelting hellstone bars without having killed a single boss.
truffle worms
The faster swing gives a Truffle Worm less time to burrow away from you, which is often the whole difference between a catch and watching your Duke fight walk into the ground.