What are their real differences? I mean I know that powers are granted and spells are taught, but more lore wise. All magic comes from the sun and stars right? but why does magic manifest into powers and spells and how does it work exactly?

I'd assume that the only difference is that you don't need great magical abilities to use powers, while you do need some prowess with magic to cast spells.

you soul is made of magic, and you utilize its raw potential to make something happen, and your soul is replenished by the holes to Aetherius. The way how can be drastically different based on which spell, which is why the spells are grouped the way they are: it require differnt ways of thought to do some spells rather than other.

As far as I can tell, powers represent a race's talents and abilities. Redguards are more warlike and are able to engage in more reverent battle. Dunmer are more religious, and tend to keep their ancestors around as guardians. Imperials are good diplomats. These attributes manifest in game as the powers of Adrenaline Rush, Ancestor Guardian, and Voice of the Emperor (most of them are even named logically). The birthsign powers work in the same fashion - people born under a certain stellar configuration tend to possess certain attributes. Can't really speak to the ones given by the stones - my guess is that they are some sort of magic learned by interacting with the stone.

One thing that came to mind today was if magic is stronger when the sun's shining.

Does seem logical, doesn't it? But I can't say I've noticed any difference when playing the games... Would be cool if they implemented it somehow - or explain why it isn't so...
Maybe magicka isn't hindered by Nirn's mass or something - so it passes right through.

Actually the magic from the Sun is not as pure as the magic from the stars, so it is other way around, which is the reason why Ayleid wells are recharged at midnight (see http://www.imperial-library.info/content/oblivion-magic-sky)

I figure that the main difference is that powers are natural abilities of one's race or birhsign, and one does not need special skills to use them. Spells, on the other hand, must be learned through study and practice,

I'd also say that some powers in the game probably aren't magic at all but are just implemented as something that works similarly to spells due to gameplay mechanics. Take the Khajiit Eye of Night power, for example. In game, it behaves pretty much like a generic nighteye spell but doesn't require any skill to cast, but somehow, I doubt Khajiit are actually all born with the innate ability to cast a nighteye spell. It seems much more reasonable to assume that what it actually represents is an ability to see in low light conditions superior to what other races have. Bethesda probably just figured that it would be annoying to have the nighteye effect constantly active even when you don't need it, especially in Oblivion where nighteye actually adds a special shader to the screen instead of just increasing the brightness. Now, of course, some powers don't seem like they can be explained by mundane means, for these, but they're still probably some sort of innate thing.

ya, magic is often used in TES as a way around game mechanics: remember "building" a dwemer Centurion IN morrowind? given, you could summon an existing centurion, but unless you built one in the first place it would attack you also.

Probably the same deal of how necromancy is handled. Collecting bone fragments, flesh, and ectoplasm in order to create a skelly, zombie or ghost, respectively, would probably spell too much work than just summoning a daedroth from Oblivion, whom are more readily available to summon from the planes of Oblivion.

We haven't been able to do conjuration necromancy yet, summoning undead is actually summoning them from a different place (which is why the mages guild teaches it in Morrowind and Post-reforms.)

[Redundant

Given how Greater Powers work, they seem to me like a throwback to DnD.

A Spell is something a person has control over and casts themselves.

A Power is more passive. It's something you're blessed with, either through race, birth sign, or in some cases blessings by a deity, shrine or powerful magic user. Enchantments on armor or clothing items could count as a form of very minor power, as long as you're wearing those items.