In Idle mode, the oscillator is still running (so the RTC and timers will keep doing their thing), but the CPU core is shut down until an interrupt arrives. You can set a timer, or the RTC alarm interrupt, to wake the CPU at a chosen time.
In Power Down mode, the oscillator is shut down too. Only an external interrupt or reset can wake the CPU up, since the RTC and timers aren't running without a clock.
Sadly, this means that going into Power Down mode makes the time kept by the RTC useless, as discussed in LPC21xx/RTC.