A: Because the writers wanted to give Bond style.
You asked for the “science”, and the answers so far reflect that. To me this misses the point entirely. I’m interested in the artistic decisions, as they are more relevant to the movie series:
1.) Cocktails are all about taste and style, (not unlike print typeface.) Bond drinks the most debonair cocktail - a Martini. It’s the mixed drink equivalent of a black tuxedo or an Aston Martin DB5. (Albeit a martini is a distinctly American invention.)
2.) However, James prefers Vodka. This implies an affiliation and taste for things Russian – not insignificant at the height of the Cold War to have a British agent prefer Russian vodka to British gin. In the sixties that’s a big statement; and in the 60’s they didn’t have a vodka from every nation like it seems today.
3.) ‘Shaken not stirred’ underlines further Bond’s rapscallion nature. To my mind, this is way more significant artistically than any molecular scientific understanding of the differences.
If the gag was Bond wants a cold but unbruised drink, they might have just chilled the glasses and the vermouth better. Also, looking at the YouTube link in the OP, I was surprised to see that Bond makes an uncharacteristic mistake (read: the writers – twice!) A martini is a martini. A vodka martini is a vodka martini. You do not ask for, “a martini” and expect, a.) there to be vodka in it, and b.) a reputable bartender to ask you if you want vodka in it. The correct drink request is “vodka martini”, then qualified simply “shaken, [not stirred].”
And, no, a chocolate martini is not a martini – its’ a chocolate martini. A “martini” with zero vermouth is not a martini – it’s gin. (Hence all the whimsical jokes about minimal vermouth (by “just waving a bottle over it”) for a very dry martini – vs. just a “no vermouth please”.) It’s a kind of unwritten but very traditional and long standing Reinheitsgebot law for martinis. A martini is: gin, dry vermouth, an olive or lemon twist garnish. Full-stop. (There is nothing ‘rapscallion’ about not knowing how to order a drink correctly.)
- I read somewhere the liquor used for Bond films was actually ginger ale for champagne, watered down coffee for whiskey, and pure water for Bond's trademarked vodka martini.
- In You Only Live Twice, Mr. Henderson gets the line backwards but they keep it in the film. He says “Stirred not Shaken.” (Also, wearing his slippers on the tatami mats is lame. [Bond is in just socks.])
- Not about drink, but this song is an all-time cine favorite (pretty great movie too!). But when I think of Bond movies I often pine on the opportunities where that 1971 Ennio Morricone song could have been used. (The song name is #21 - Tarantola Dal Ventre Nero The connection is that the original movie has two Bond girls in it.)
These are just hard won thoughts. Nothing to back me up online that I know of. Necropost. Sorry.