I'm not sure that a "concrete example" of mindless activity vs. mindful activity is possible, because it requires the use of subjective interpretations of what constitutes either of those two notions.
Mindfulness is a state of active, open attention on the present. When you're mindful, you observe your thoughts and feelings from a distance, without judging them good or bad. Instead of letting your life pass you by, mindfulness means living in the moment and awakening to experience.
Using this definition, it seems as though there are relatively few activities one can perform while being mindful. If an activity requires a high level of concentration to perform, than it would be difficult or impossible to simultaneously focus on the meditative internal and external reflection implied by 'mindfulness.'
Examples of some activities that might be more favorable to such meditation:
- Going for a walk or hike somewhere
- Going to a quiet, secluded location
- Doing chores around the house that require little concentration
Examples of some activities that might not be favorable to such meditation:
- Working on a task that requires a lot of concentration or creative problem-solving
- Watching an interesting movie or reading an interesting book, (unless perhaps you pause for self-reflection from time to time.)
This is also tricky because it may mean different things to different people. Mindlessness might mean simply the opposite of mindfulness, in which case the counter-examples I listed above might be considered 'mindless.'
However, it seems a bit strange to include things like creative problem-solving and socializing as a "mindless" activity; because your mind is absolutely engaged in what's going on, just not in the same sense as it is for a mindful / introspective reflection.
Perhaps a fair definition of "mindlessness" is if you're by yourself with some free time; instead of choosing to engage in some social and/or mental activity, or meditate in a mindful way, you just sit around; being bored, thinking about how boring something is, etc.
Yet even "mindlessness" with it's negative connotations may have some importance. Sometimes when you're undergoing great stress, mindful meditation may be good and useful but you're not ready to do it; you may need to distract yourself from something that's troubling you. You might have to try something "mindless" to clear your head a bit before you can even be "mindful."
The truth is; the human brain never truly slows down while you're alive. There's a persistent myth that we only use 10% of our brain; but while it's true that only portions of our brain are active at any given moment, we actually use 100% of our brain, just not all at the same time. It would be destructive and/or impossible to use 100% of your brain at once, that's not how the brain is designed to work. http://ed.ted.com/lessons/what-percentage-of-your-brain-do-you-use-richard-e-cytowic
In this sense, the question of "mindfulness vs. mindlessness" isn't necessarily about whether or not you're using your mind. You use your mind regardless, even in your sleep. The question becomes whether or not you're using your mind well, and that ultimately is a personal, subjective question.
The closest I can say to understanding it objectively is that it's probably healthiest to strive for balance in how you live your life.