Tubeless vs Tubeless ready I've decided to upgrade to road tubeless. Hutchinson sells both "Tubeless" and "Tubeless Ready" tires. According to Hutchinson' website, Tubeless Ready means that (in addition to having a special bead) the tire's tread is air permeable and will gradually lose air, so sealant is needed, not just to seal punctures, but to slow the loss of air. 

Their Tubeless tires, on this other hand, have a butyl lining that prevents air loss. Although a sealant can be used to seal punctures, it's not required.

My question: Do tubeless tires -- the tires that have a lining -- hold air as well as tires with tubes? (Note, my car's tires don't have tubes and they hold air quite well.) I ask because I need my tires to stay pumped up, and not go soft overnight the way they usually do on my mountain bike.
Proper tubeless (UST) tyres do indeed hold air well, but are also heavy and don't roll as well (due to extra material that needs to deform around objects).  Due to the nature of MTB'ing you are also likely to need sealant for puncture protection, as whilst UST tyres are more robust, they can't shrug off all potential punctures.

This is why you see very few UST tyres available for sale now, with most riders (and manufacturers) preferring tubeless ready.

In my experience, a well set up tubeless ready tyre with sealant holds air at least as well as an old tyre/tube combo.  I typically lose ~2psi/wk on my tubeless ready setups so usually top them up once a week, but could go 2 weeks if necessary.