The advantage of powered monitors is that the internal power amp provided by the manufacturer is already matched to the speakers. This means you don't have to audition a bunch of power amps yourself to see which one works best with your particular monitors. It also means less cabling (just a line level into each monitor - no speaker cables necessary).
The advantage to a seperate amp/speaker system is mostly if you already have a power amp, non-powered monitors are usually cheaper. Also, one amp can be used to power several sets of monitors (with a switcher).
In general, for home studio use, if you are starting from scratch, I recommend getting powered monitors. If you don't have a power amp already, given your budget, by the time you get a good one there won't be enough money left over for speakers.