That seems to be the problem of the original poster. Each of the power sources in 4e represented different kinds of energy and forces in the universe that you could use to do stuff. It's different to get a fireball by creating it using arcane power, than getting one by praying to the gods for it. Yeah, you may have got the same fireball in both actions, but how you got it also matters.
The thing to remember about the cognitive gap is that it's the area between what the game's mechanical operations perform, and what the game tells us those operations mean in the context of the game world. In that sense, the power sources "matter" (where "matter" is taken to mean "helps us bridge the gap") where the game presents them as being mechanically distinct from each other, and gives in-character weight to what those distinctions are indicative of.
Now, "power sources" have been part of D&D since the beginning, e.g. arcane and divine magic. But while different editions have lent different weight to them, there was typically always some degree of differentiation between them (even if a lot of it was folded into the classes that utilized them). Divine magic required you to adhere to the tenets of your faith, leaving you subject not only to divine review of your conduct (potentially losing spell access if your deity didn't like how you were behaving), but also had you
requesting spells via prayer instead of just preparing them without overview; your god might very well not give you the spells you asked for, but something else altogether (which was supposed to be the DM's way of helping you out for an upcoming adventure, at least as I read it).
Arcane spells, by contrast, were things you either had to find or invent on your own. You needed to record them in a spellbook, which could be lost or stolen or destroyed (and so you better have backups prepared), and even when you found a new one, you weren't guaranteed to be able to learn it. Specializing in a school a magic was something you could do to help out with that, and make your spells harder to resist.
And that's not even getting into things like psionics, which had very different mechanics from arcane or divine magic. Or incarnum in 3E, which was its own system. Or spellfire, etc.
In standardizing a lot of the mechanics related to those, 4E widened the cognitive gap, doing away with many of the details that differentiated things. There's no check or roll involved with wizards learning new powers, and clerics are explicitly stated not to receive their spells from their god. Psionics uses AEDU just like arcane and divine magic. And that wasn't just on the mechanical side, either; the "martial" power source talks about what it does, but not what it
is, etc. There's a reason why "sameyness" is a term that gets thrown around in this regard. Certain things remained (e.g. wizards' spellbooks), but the formalization of "power sources" as a term under the rules didn't mean that it hadn't reduced their distinctiveness in how those rules operated.
Now, it should be noted that 4E isn't unique in having this particular aspect of the cognitive gap, but overall it did widen it. 3E had problems in terms of explicitly allowing for clerics with no gods, doing away (as of 3.5) with psionic combat, allowing arcane spellcasters (such as bards) to cast healing spells, etc. 4E simply pushed the envelope in that regard, and for many people that widened the gap to the point where bridging it was more work than they were comfortable with.