Shaken, not Stirred

I dont know about A PINK one....but I do know that most bartenders and martini drinkers perfer the drink stirred not shaken as it supposedly 'bruises' the gin if you shake it...which is why Bond always specifies that he wants his Shaken And Not Stirred...it sticks out as unusual...like the man himself

If it's pink, it's not a real martini.

To answer your question, the 'experience' is in the eyes of the beholder, so to speak.
Make two cocktails, one stirred and one shaken, and do a blind taste test.
Betcha can't tell the difference.

this is from Wiki, explaining what bruising is and how it is thought to alter the flavour of a martini:

The term comes from an older argument over whether or not to bruise the mint in preparing a Mint Julep. A shaken martini is different from stirred for a few reasons. The shaking action breaks up the ice and adds more water, slightly weakening the drink but also altering the taste. Some would say the shaken martini has a "more rounded" taste. Others, usually citing obscure scientific studies, say that shaking causes more of a certain class of molecules (aldehydes) to bond with oxygen, resulting in a "sharper" taste. Shaking also adds tiny air bubbles, which can lead to a cloudy drink instead of a clear one. If the drink is used as an aperitif, to cleanse the mouth before eating, the tiny air bubbles restrict the gin (or vodka) from reaching all tastebuds. This is why purists would claim that a martini should always be stirred. Some martini devotees believe the vermouth is more evenly distributed by shaking, which can alter the flavor and texture of the beverage as well. In some places, a shaken martini is referred to as a "martini James Bond" or a "007." (Fleming actually named Bond's drink the "Vesper", after the heroine of the first novel Casino Royale, though it is a specific recipe.) In a scientific study, researchers with the Department of Biochemistry, Faculty of Medicine and Dentistry, University of Western Ontario, Canada, determined in 1999 that a shaken martini is demonstrably more healthy than a stirred one. Antioxidants are known to promote health, particularly by reducing the incidence of such age-related diseases as cardiovascular disease, stroke, and cataracts. Antioxidant properties are possessed by alcoholic beverages in general, including martinis; but in carefully controlled tests, the researchers determined that a shaken martini has significantly higher antioxidant properties than a stirred one. As they humorously concluded in publishing their results in the British Medical Journal, "007's profound state of health may be due, at least in part, to compliant bartenders."[9]

Small point of interest, the technical name for a martini that's shaken rather than stirred is a 'Bradford'.