Well the difference between deit and sid is:
Deity def units to start 8, sid is 12.
Deity off units to start 4, sid 6.
Here is another big one:
Deity 1 extra settler, sid 2 extra ones, so they start with 3 settlers.
Deity 2 workers, sid 4 extra ones (beyond what all levels get).
Free unit support Deity 16, sid 24.
Cost factor deity 6, sid 4. So at sid they get 100 shield items for 40 shields. 400 shield wonders cost 160.
Just a few of the bonus changes. Now as to Regent, it is straight up no bonus for the AI.
Monarch they get a 10% bonus, so they pay 90 shields, you pay 100. They get no extra workers or settlers, but start with 2 def and 1 off units.
These units are the best they have tech for, so most will be warriors, but some (Germans) start with archers and spears, so do not try to rush them.