+5SB is on whenever your PSU is not switched "hard off", not just when the PC is "soft off", so just wiring an LED to +5SB and ground won't give you the effect you want.
The conceptually easy and foolproof (but a bit clunky) way to have something (like your LED) turned off when something else (like your main power supply outputs) is turned on is with a relay with "normally-closed" contacts. A relay is just a switch that's flipped when current passes through a coil - an electromagnet. "Normally closed" means that the contacts conduct electricity when the relay coil is not energized. You can use one of the 12 volt "Bosch relays" for this. These are very easy to find in auto parts stores and are often used by car modders and stereo installers. (It is a convenient accident for us that 12 volts is available both in cars and in PCs.)
Just be sure that the relay you get has "normally closed" as well as "normally open" contacts, i.e. it's a SPDT (single pole double-throw) type.
The relay will have quarter-inch quick-disconnect terminals with small numbers next to each. Connect the relay coil (terminals 85 and 86) across the PSU's +12 (yellow) and ground (black) wires. (If you have a spare four-pin "Molex" drive power connector, that is a great place to connect to those.)
Wire the normally closed and common contacts (87A and 30, respectively) in series with +5SB, your LED, resistor (680 ohms is probably good), and ground. The anode of the LED goes to the +5SB side of that path, the cathode goes to the ground side. Easy.
Just, as someone else said, do not cut the +5SB wire (at least not permanently). Splice another wire into it instead. If you don't want to solder, the auto parts store will also have 3M "Scotchlock" insulation-displacement connectors that will let you splice into that wire easily.
Either way, be sure the PSU is completely off while you do the work. Either pull the AC power plug from the wall, or, if the PSU has a hard power switch on its back, turn that off.
There are other ways to do this. A single, very cheap IC chip like a 74AC00 could be used. Or a couple of transistors.
There is a trick involving using the PC's main +5 supply as ground for the LED, strange though that sounds. This lets you get away with nothing but the LED and resistor. 150 ohms would be about right in this case.
But the relay is the most straightforward and works in the most intuitive way.