Vodka and gin both begin as a fermented grain or fruit "beer" or "wine" that is then distilled into a higher proof neutral spirit. From there, the differences happen right away.
Both tradition and production regulations — from distillation techniques to the addition of botanicals — aim to give gin distinctive character, aromas, and flavors.
In the United States, vodka is defined as a neutral spirit, distilled at or above 95% alcohol by volume (ABV), or about 190 proof, and made from any material.
The thing that makes gin what it is are the botanicals, which can include seeds, leaves, stems, spices, and citrus peels. The most important ingredient is the juniper berry.
The botanicals may be introduced into the fermented mash bill and distilled along with the initial "beer," or added to a distilled neutral spirit (vodka) to macerate or soak.
In cocktails, gin will generally be more full-bodied, aromatic, viscous, and flavorful, while vodka adds structure without overly influencing the flavor of the drink.