Twice as much information.

My go-to for I/Q is: Having two allows you to represent negative frequencies. With a normal, real signal, this is of course impossible (negative frequencies will automatically mirror the positive ones), but if you have a signal centered around e.g. 1 MHz, there's room for above-1MHz and below-1MHz to be meaningfully different. And _that_ allows you to get a complex signal (I/Q), once you pull the center down to 0 Hz for convenience of calculation.