According to some, aerating the gin by splashing it around alters the flavor, or "bruises" it; or more perceptibly, one would be likely to quickly notice the little bubbles disrupt its normal tongue/palate texture.
From Wikipedia, "Shaken, not stirred":
Some connoisseurs believe that shaking gin is a faux pas, supposedly
because the shaking "bruises" the gin (a term referring to a slight
bitter taste that can allegedly occur when gin is shaken). In
Fleming's novel Casino Royale, it is stated that Bond "watched as the
deep glass became frosted with the pale golden drink, slightly aerated
by the bruising of the shaker," suggesting that Bond was requesting it
shaken because of the vodka it contained. Prior to the 1960s, vodka
was, for the most part, refined from potatoes (usually cheaper
brands). This element made the vodka oily. To disperse the oil, Bond
ordered his martinis shaken; thus, in the same scene where he orders
the martini, he tells the barman about how vodka made from grain
rather than potatoes makes his drink even better.
Better, from the Martini FAQ;
...while we should all defer to the inimitable Mr. Bond on matters such
as high-tech spy gadgets, impromptu hand-to-hand combat, and
retrograde seduction techniques, his reasoning on this matter is
specious. To "bruise" a wine or spirit means to take some action that
changes its taste. Agitating and therefore aerating a gin or vodka
martini changes its taste: it makes it taste "sharper". It imparts a
certain bite or zing. Given this, and given his dislike of a bruised
spirit, Bond should insist that his drink be "Stirred, not shaken,"
since shaking "bruises" the gin more than stirring does.
...Here are the facts: Shaking cools a drink more quickly. Shaking is
more likely to chip small shards off the ice, some of which will make
their way into the drink, no matter how carefully one strains and
pours. Which may be part of the reason why... Although the gin spends
less time with the ice when the drink is shaken, shaking a drink
actually dilutes it more than stirring does. Very rarely, shaking can
produce a chill haze (the precipitation of very small solid particles)
from the vermouth, giving the drink a cloudy appearance. Shaking
creates tiny bubbles in the mix, which temporarily impart a cloudy
appearance to the drink. Shaking causes a certain class of molecules
in the liquor (aldehydes) to combine with oxygen more than stirring
does. The oxidation of these molecules also slightly alters the
flavor, making it "sharper" (Miller, Anistatia R. and Jared M. Brown. Shaken Not Stirred: A Celebration of the Martini. New York: HarperCollins, 1997).