Both methods are used to remove dissolved solids (ions) from water. In basic terms, the distillation process boils the water in a still, and the steam is condensed out and collected. Any of the non-volatile components that were in the water are left behind, so you end up with water whose mineral content is usually less than 10 parts per million.
Reverse Osmosis forces water through a filtering membrane. The pores in the membrane are small enough to filter out a lot of the molecules that make up dissolved minerals, but it can't filter out all of them.
Between the two methods, distillation will give you water with the least amount of dissolved solids.
As far as using either type on orchids, there's a huge debate about whether it's really necessary if you have good quality tap water.
If you decide to try one or the other, you may have to add back many of the dissolved minerals (Calcium, Magnesium, Potassium) that were taken out: orchids (and a whole bunch of other living things) need them in trace amounts for sustenance. Fertilizers that contain trace elements will more than likely take care of that.
Our tap water here is very good, so I don't use any kind of filtration before the tap water hits the plants. So far, I haven't had any problems.