For me, the difference is this...I'm not actually free.
I have a job, and it's a job I love most of the time, but its' still a job.
This.
My DH is RE. He worked full time out-of-the-home for many years, and left when he qualified for full pension benefits. He did not seek (or find, or take) another position of paid employment.
We have a kid, who is now in elementary school. In addition to public school, we do make use of a lot of (paid for) organized kids' activities, because we are fortunate to have a bunch of great ones around us and frankly, they do a much better job (and involve a lot more kids, which DS appreciates) than DH (or I) would of engaging DS's interests and providing him with opportunities to do new things.
We also do a lot of stuff with DS ourselves -- me in the time I'm not at my office/working, and DH other times (or all 3 of us together). And we have a wonderful in-town grandparent who typically does stuff with DS for 4-8 hours per week without our involvement.
And DH is the "fall back" for caring for DS, and of course has been called into service a bunch lately, what with all the snow days.
I often don't call him a SAHP as that isn't really what he wants his role to be and indeed there are many ways in which our household is organized that are more similar to those in which both parents are WOH than not (see above). So I tend to call him a SAHH or RE. But even with that, what he does is a big job and not one whose importance I downplay, or whose ability to generate exhaustion I underestimate.
(Another non-fan of the "domestic engineer" label here, though I did appreciate the one of my mom's friends who, feeling similarly annoyed at people's reaction to the "SAHM" or "housewife" label in an earlier era would tell people that she ran "a home-based injury prevention program")