They are not the same.
Distilled water has basically zero minerals because it is made by boiling water and condensing the steam. It also has zero buffering capacity for pH and can quickly become acidic once exposed to air.
RO water has some minerals in it and is much better to use for watering plants.
RO water should be cheaper is you have to buy it.
The residual dissolved minerals in the RO water is dependent upon several factors, with the top four being:
TDS of the incoming water supply
What those minerals are
Age of the membrane
Design of the membrane.
Basically, the membranes are designed to reject X% of the incoming dissolved solids. Most of those we see are in the 98%-99% range, although it varies from 93% up, depending upon the intended application. Even that is more than sufficient for orchids, though - If we use the excellent NYC water as a "benchmark" at 40-50 ppm, that "only 93%" membrane could match that with in incoming supply in the 600-700 ppm range!
In biology lab (we do DNA and some RNA works), we usually don't use distilled water. We call it ddH2O (double distilled), but in reality, it is deionized water (we use Mili-Q for lab works). Here is some info about different methods of water purification: http://en.wikipedia.org/wiki/Purified_water
In practice (for growing plants), RO should work as others said.
I have some customers that raise dendrobates (poison dart frogs), and a few others with marine tanks, and they are all very keen on very pure water, and they simply attach a DI column after the RO. That is probably as close as anyone will come to distilled.