It depends, but mostly only for specific ingredients and mixtures.
Some minor changes in how much ice will melt, and thus a shaken drink will be slightly more watered down. In a drink like a martini it would be very difficult for most people to tell the difference between shaken and stirred.
If the drink includes an egg white or some other protein based ingredient then a vigorous shaking will cause it to foam up nicely and impart a rich creamy texture to your drink. A stirred whiskey sour versus a properly shaken one will both look and taste very different.
Some thick syrups and fruit juices can be shaken to get a thin froth on the top of the glass which makes a small difference over all but can be very pleasant if you personally enjoy that little extra fluff as you sip a fresh drink.
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