There are other Dominion effects now besides the morale adjustment. What you get depends on who you are and how you are playing.
Your Dominion will adjust many facets of a province, from productivity to the birth/death rate. The "scales" you adjust when you design your god define a lot of the nature of your empire. You can be a lucky, magical nation of lazy slobs, or a bustling, severe people who have little time for magic. I don't know how much of that was in Dom I, but I know it makes a major difference in Dom II.
The themes, which are really variations on the god's Dominion, have a wide array of effects. There are ones available to all nations and ones that are nation-specific. Some, like Pangaea's Carrion Woods dominion/theme, will depopulate a land of those pesky humans pretty quickly. And that's OK: the dead ones are good for making into Manikins.
Finally, some spells/abilities are only usable within a positive Dominion. For example, an immortal unit is only immortal within its diety's realm of influence! Also, a spell like the Eyes of God is a global enchantment, but the level of potency it has depends on your Dominion in a given province: if you dominate the place, you see everything; if some other god has them bowing and scraping, then things are a bit more cloudy.
I'm probably leaving out stuff . . . I've got this feeling I'm neglecting something important, but I can't think what it could be right now.
