In the context of family life "doing" the dishes implies that you will deal with the dishes. This may mean any (or more likely all) of the following:
- rinsing and putting dishes in the dishwasher. It may also imply putting the dishes away after dishwasher is done. But that is two hours later, I am a male, and any grey area is a negative to me.
- washing the dishes, drying the dishes, and putting them away.
- putting away already cleaned dishes
Now washing the dishes means that you scrub the dishes with a pad, soap, and water. This means something different to me. If I told someone to wash the dishes I wouldn't want them loading the dishwasher either.
So if you tell someone to "do the dishes" you are giving them free reign to deal with the dishes but you are also implying they have more tasks than simply wash the dishes.
If you tell someone to wash the dishes, they might comment back, "OK but you are drying."