The Internet has not changed the way I think. The old stone-age mental software still seems to be working surprisingly well in the 21st century, despite claims to the contrary. What the Internet has done, however, is sharpen my memory.
A quick search with a few well chosen keywords is usually enough to turn a decaying memory of a half-forgotten article, scientific paper or news item into perfect recall of the information in question. Previously, these things at the penumbra of recollection could only be recovered with a great deal of effort or luck. The Internet has, in effect, upgraded my memory of such marginal items from haphazard and partial to reliable and total. This means I can swim freely through the Internet's vast oceans of information, safe in the knowledge that any connections between items that subsequently occur to me can still be made. (My own work as a journalist and author is based on making connections in this way, but the same is true for many other information workers, a category that encompasses a growing fraction of the workforce.)
This is useful now, but I expect it to become much more useful as I get older and my memory starts to become less reliable — moving more of the information that passes through my mind into that penumbral region. Indeed, I am reminded of the impact that eyeglasses had after their development in the late 13th century (though my recollection of the details was sketchy until I, ahem, asked the Internet).
As Giordano of Pisa noted in 1306, "It is not twenty years since there was discovered the art of making spectacles that help one see well, an art that is one of the best and most necessary in the world." Eyeglasses doubled the useful working life of scribes and skilled craftsmen who were otherwise liable to suffer from farsightedness (presbyopia) from the age of around 40. The historian David Landes has suggested that this use of technology overcame what had previously been regarded as an unavoidable human limitation then spurred further innovations of a similar nature, such as the development of fine optical instruments and precision machine tools.
Perhaps the same will be true of the way the Internet enhances our mental faculties in the years to come.