Giving up and moving on are two different actions. It is possible to give up on a task, and not move on for fear of encountering similar difficulties.
However, it is also possible (in certain situations) to move on to a new task without giving up on the previous one. This would imply that you intend on eventually returning to the first task to complete it at a later date.
The two actions are related, but they are not identical in their function or consequences.