There is a difference in taste between a Martini that is shaken and one that is stirred.
Shaking a Martini will crush the ice that has been put into the cocktail shaker with it and as a result the drink will appear cloudy. The ice will melt more rapidly and affects the taste. Stirring the Martini doesn't break the ice up and the drink will appear clear. if you line the two up and taste them, it will be different
Once the drink has been shaken, stirring will not make a difference