The difference is only a theoretical though, you would not be able to circumvent taxes by saying I'm not getting paid, I am only compensated with a Bank account which has a limit of $x every month. So yeah, there is a theoretical difference but not a practical one.

But the argumentation is kinda strange, if you think you are not compensated correctly you could always get a job and leave child/elder care to somebody else. If you don't want that, it seems that you are ok with the compensation. If your partner does not want that, the problem is not the unpaid work but the gender roles. If you can't get a job which pays more that you would spend, one could argue that you are probably compensated fairly.

> A woman that loses her home because her husband gets fired from his job now has no compensation through no fault of her own.

This is imo a valid point, because she would still have to do the work without compensation (even though as both don't have paid work they would hopefully share the work)