If you are a man, the hot water has a deleterious effect on your testicles' ability to make sperm. But so do saunas apparently.

https://pubmed.ncbi.nlm.nih.gov/23411620/

That one was 80-90C, which is a really hot sauna.

Yes, if by hot bath you mean submerging yourself to neck level in 40ºC or above water for 20-30 minutes. There's no reason to believe any "heat therapy" modality is superior to another as long as you suffer equal heat stress.

For the record, if you're not acclimated, intense heat exposure is a lot more agonising than 30 minutes of exercise for less benefit. If you haven't experienced a properly tuned sauna in your life you are in for a ride. What's being studied in the literature is nothing like your standard hotel experience.

It might not do the exact same, but it will have some effect. A lot of the benefit comes from the raised heart rate and opening of the blood vessels that the sauna produces, and I can expect that a warm bath would also have a similar effect. I think both are also known to reduce stress, which can help to lower blood pressure.

Almost certainly but most people don’t find it as enjoyable. Also the problem of keeping the bath hot enough for 20-30 mins.