I don't understand the difference either other than Assistant seems to be replacing Now. The issue, I'm finding, is that Google Assistant requires that you turn on the recording of usage data and history to your account, such as Web & App activity, Location History, Voice & Audio activity. All recorded and tied to your account. I don't believe this was the case with Google Now.
Google, to their credit, normally allows you to erase all this data and turn recording off, but with Google Assistant, they require it all to be on and recording. I avoid using it because of this restriction.
IIRC one could use Google Now with voice recordings off and the extra-spying setting of "web and app activity" off (though web and app activity had to be on for many features). All this must now be on for the assistant, as you say.
I did try out Google Now for a time due to having an Android Wear watch and found that as time went on more functionality required the turning on of these settings. One particular oddity for a time was that "OK Google, Navigate Home" would produce only a complaint about web and app activity being off, but "OK Google, navigate to $HOME_STREET, $HOME_TOWN" was fine.
Google, to their credit, normally allows you to erase all this data and turn recording off, but with Google Assistant, they require it all to be on and recording. I avoid using it because of this restriction.