Chronicle Careers has a very good article about what it looks like to do intellectual work.
As a sort of intellectual myself, having trained in and having worked in the field of law, where much intellectual work goes on, I can tell you this:
Most people would look at me and think that I'm doing nothing. Being reared in a family that regards activity of some sort as the measure of work has made my life difficult at times.
Frequently, I've been accused of being lazy or worse yet, available to assist with whatever task needs attending to, when I've actually been involved in some intellectual work.
It's not uncommon for those around me to accuse me of slacking off when I've actually been heavily engaged in important intellectual pursuits. Somehow, the anti-intellectualism common to the South encourages disdain for eggheads.
Too bad. I've weathered the criticisms.
Too often I find my intellectual work de-valued. No more.