Godhood is decided primarily based on power, which in turn is generally decided based on the number of worshipers the prospective deity has (paraphrased from DMG 5e, p 11 under Divine Rank).
That means that gods - both lesser and greater - grant spells to clerics as a defining trait. Not being able to grant spells to clerics makes immensely powerful beings quasi-dieties of some sort.
In FR the dragons were created by the primordials. Does this mean many/most/all dragon gods are also primordials, as aspects/fragments of Io?
The Dragon Gods are presented as actual gods rather than primordials (DMG p. 10, PHB p. 296, & SCAG p. 113).
Can a primordial also be (or somehow become) a god?
I'm unaware of this occurring, but I wouldn't rule it out:
- Asgorath/Io is both, but (as creator of the universe) doesn't count.
- Tharizdun is close, but is not a Primordial.
- Kossuth is also close, but he's "not a true god but actually an
elemental primordial".
Do primordials need, desire or benefit from followers?
Not normally, but Kossuth is an example of a primordial that benefits from extensive worship.
Do surviving primordials have significant religions associated with them, either in Abeir or in Toril?
On Toril the five Elemental Lords have followings, while The Seven Lost Gods were once worshiped. I believe Abeir would have more significant Primodial worship, but that setting was never detailed.
Do primordials grant spells in the same way as gods?
Not normally, but again: Kossuth is an example of a primordial that both grants spells and has extensive followings.
Can they be warlock patrons?
I don't see anything preventing this. A few homebrewed examples of this can be found here and here.
Is there any obvious difference in the portfolios of primordials versus gods? I would have assumed that since they predate mortals they tend to represent more fundamental forces (like the elements) rather than human concepts like law, love, luck, agriculture.
That might be one way to view it, but 4e's creation myth (Worlds and Monsters p.56, cited below) has the Gods being created at the same time as the Primordials. Another way to view it might be that the Primordials are composed of elemental "physical-matter" while the Gods (and Astral Sea) are more composed of thought (mental-matter).
In short, what is the difference between a god and a primordial? Are they fundamentally the same (just different lineages) or are there fundamental differences?
They are consistently presented as having fundamental differences. From 4e's Worlds and Monsters (p.56): "The gods, beings of divine power, appeared in the Astral Sea, while in the Elemental Chaos arose the primordials, incarnations of tremendous elemental might"... "composed partially of creation-stuff".
I also found this fantastically detailed post on Candlekeep...
That still seems like as good a guide as any.
Best Answer
In 5th Edition, demigods are the offspring of deities & mortals that fall under the quasi-deity category (though they are considered the least powerful quasi-deity). Quasi-deities are beings with divine origins which, in contrast to 'true' greater/lesser deities, have insufficient power to hear/answer prayers, grant cleric spells, or control aspects of mortal life (though quasi-deities have the capacity to become full gods with sufficient worshippers).
(Source: 'Divine Rank' sidebar, 5e DMG p.11)
I do not have enough experience with older editions to contrast against their definitions of demigods vs 'true' deities.