Shaken Versus Stirred
The Eternal Question and Some Guidance on The Subject
First why does this question exist?
In my opinion (most of this is opinion by the way outside of the physics) you can blame Ian Fleming and the Bartender who popularized the shaken martini.
Prior to this it was not done except by the unknowledgeable. As a rule, clear, uncarbonated drinks were stirred, juice based (not clear), non carbonated, were shaken and carbonated drinks were built.There were, and are exceptions but these are the general rules.
This mostly had to do with the physics and serving of a drink. "Bruising" the gin or vermouth is a misnomer. It has much more to do aeration,dilution and chilling dynamics than traumatizing the ingredients (although bruising is a evocative shorthand).
When you stir a drink, you are mixing the ingredients gently and minimizing the amount of aeration (think of pond or aquarium turbulence to infuse oxygen) in a drink and minimizing dilution through mechanical vibration.
Bottom line, is you get a very chilled limpid, almost liquid glass appearance to the drink, and a much thicker body, texture and mouthfeel.
Shaking, on the other hand, while efficient at mixing vastly different (by appearance, specific gravity and other factors) ingredients oxygenates the drink,introduces turbidity as a result and the mechanical agitation increase dilution (25% after 20 seconds of shaking on average- thanks to Gary Regan for the research!)
The same principles apply to and perhaps a more common (or extreme) example is the regular cocktail (on the rocks or especially straight up) compared to a frozen, blended version. Notice the difference in appearance texture and taste between a a frozen margarita or Daiquiri and one prepared in the more traditional manner.
All this being said, it comes down to personal taste. As Gary Regan teaches in his Cocktails in the Country course, "There are No Rules" (except, of course, that one) and "Nothing is Written in Stone."