Alright, lots of misunderstanding there.
1, distilled == boiled water?
No, distilled, as ZooKeeper said above, involve turning the water into stream, then condense back into liquid. This is where it costs you: it takes lots of energy to turn water into stream.
2, distill and copper?
Many modern distillation plant use aluminum condensor due to weight and cost, but the post-distillation process contamination is difficult to control, and this and the cost issue is what make many of us use RO/DI instead.
3, Distillation == puriet water there is...
No, distillation is one of the best method to purify water contaminated with refractory minerals, but rather it is the best method (in term of purity, disregarding the cost) depends on the source water. Volatile compounds for example, will be carried by the distillation process.
Depends on your access to filtered water and the amount of water you need, for some people, distilled water or store-brought RO is the best choice. For people who is willing to pay a inital cost to buy a RO/DI filter, and have the space for it in the house/apartment, RO/DI can be a better alternative due to convience.
BTW, welcome to RDO, Dan.