How would you explain to students the difference between drawing and constructing? "Accuracy" seems to be a go to word, but that's not really what the difference is. I want to say more, but I also don't want to lead answers, so I'll stop here. =)
I would emphasize the algorithmic nature of a construction over the ad hoc nature of a sketch/drawing. Tell your students that a construction must be accompanied by a specific sequence of steps (a "recipe", if you will) so that, in the future, anyone else could come along and follow that recipe to accomplish the relevant task, as well.
For example, the perpendicular bisector of a line segment can be constructed by following a particular recipe. (See this link for the construction and its proof.) Starting from any line segment, following that recipe will create the perpendicular bisector of that segment. However, any particular sketch/drawing of a perpendicular bisector of a line segment, without any accompanying information, is merely that: a picture of one example. The recipe is the key component of the construction, not any one example of carrying out the construction.