As a self confessed Martini nut I can tell you two things, one of whom was told to me by Allesandro, head barman at the Duke's Hotel, London where they have a special Vesper Martini that's as close to the original as you can get.

In the old days people didn't mix vodka and gin in cocktails as they were much stronger than they are now. In Casino Royale Bond acts a particularly flashy, arrogant show-off type gambler. In the early 50s when the novel is set (although please do bear in mind that the story was heavily influenced by wartime events in the Naval Intelligence Division) people never mixed Gin and Vodka in a Martini, and a Martini was always stirred with a special spoon to avoid chipping the ice.

Bond, being the flashy type ordered his Martini with Vodka, which would then need to be shaken in order to avoid the oilyness that came with the potato-based vodkas of the time as per the wikipedia article. The act of it being shaken would've aroused much interest from people who would have asked what the gentleman was drinking, to which the response would seem incredible in a kind of 'who does this guy think he is' type of way. The casino is supposed to be a high brow csaino that will have seen better days, Bond was breaking etiquette to show off how big his balls were and to attract Le Chiffre's attention.

The other thing on a more technical point is that a modern vesper cannot be made with the same ingredients as the original (as the formulas have changed). In order to get something close you need a strong but fairly easy drinking vodka (at least 43% ABV), a neutral Gin as close to 53% as possible, Lillet Blanc (a white vermouth) and either a tincture of quinine or a dash of orange bitters. My personal recommendation would be for Berry Bros No. 3 Dry London Gin, Potocki Vodka, Lillet Blanc and Orange Angostura Bitters. Stirring takes longer to cool the drink down to the right temperature (around 1-3 degrees C), whereas shaking takes considerably less time, but you'll get ice particles in the resulting drink. As the drink cools more water melts into the drink, diluting the cocktail. Thus a shaken Martini is generally slightly stronger than a stirred one. The oxygenation of the mix also gives the drink a sharpness that you don't get in a stirred martini, but most people wouldn't be able to tell the difference.

The TL;DR for all of this is, vespers should be shaken and not stirred, or poured and layered (as per the Duke's bar), no other martini should be shaken - unless it's pure Gin and Vermouth in which case it can be gently shaken to waltz time.