The Dead Past is set in a society where academic fields are heavily regulated by the government and the individual scholar is forced to specialize to get government funding. Humanity has gathered too much knowledge; scientific progress requires vast amounts of resources and individuals with heavily specialized knowledge to continue, and government intervention was the solution Asimov came up with. Humanity has just passed the point where one man can know everything about everything.
Profession is set much further in the future than The Dead Past. In it, scientific knowledge has progressed to the point where learning for oneself has become absurd. There's simply too much to know. Instead, people are “educated” instantly at certain points in their lives by interfacing directly with a computer. Newly educated adults are given highly specialized talents because it is believe a given brain is best suited to a given profession. Humanity is well beyond the point where one man can know even most things. “Education” itself has been mystified.
In both of his shorts, Asimov paints societies where the way people learn, and thus the structure of academia and the economy, is heavily influenced by the supposed necessity to retain all of the information they need access to in their own minds. Academic types must specialize in The Dead Past because it's believed their time and mental storage would be wasted otherwise. Everyone is specialized in Profession for much the same reason. Preceding both plots is the question, “What will we do when we know too much to know it all ourselves?”
How remembering is changing, or, as other authors like to put it, Google making us stupid, is not a detriment to our society, but our answer to Asimov's question. Perhaps having different intentions for his stories than my own, Asimov never considers the possibility of offloading information. His societies attempt to cram as much information as possible into the limited space in the human head, and academics and economics are profoundly influenced by this blatant short coming.
We have a different answer: externalizing information. The human mind was never meant for storage; it just doesn't handle the task well. I can only recall what I had for lunch yesterday because I have the same delicious turkey-salami-provolone-cucumber-chipotle sandwich every day. I couldn't tell you what I had for breakfast. I couldn't even tell you where I ate breakfast. I walked out of my room in the middle of the night last night, to investigate an odd noise and evict a friend who was making it rather difficult to sleep. I forgot my key.
Rather than using my mind as an organic hard drive limited by relatively slow read/write speeds, shoddy recall, and minuscule storage space, I should have used it as a processor, sifting through information stored in the much more reliable form of the note on my door (that I ignored...) saying in bold red letters “REMEMBER KEY CARD.” Had I used my door as my memory, rather than my head, I wouldn't have had to walk down and up four flights of stairs at one in the morning.
Offloading information presents its own challenges, of course. Much of the outcry seems to be from the older generation against the behavioral differences between our generation and theirs, and some cries voice well-reasoned fears, many echoing Socrates' complaints in Phaedrus. They aren't entirely wrong; there certainly will be complications, but there is no reason to fear this particular change. We know too much to continue using our minds the way we have in the past.
Now, I don't want to say Asimov got it all wrong. He's written a lot more than I have read; he may have reached the same conclusion elsewhere. But it seems we have found a better answer than Asimov ever dreamed up. Rather than using our minds as storage space, we use them as tools to sift through what we have stored elsewhere. Externalizing information allows us to spend less time memorizing, and more time thinking. It will change our society, change us, not by turning us all into pale, under-socialized basement dwellers who wouldn't recognize another human if they met one in person, but by preventing intellectual stagnation and allowing us to build a new culture centered on thinking rather than knowing.