The answer to this question depends on one’s understanding of what constitutes a zombie. According to the popular image, a zombie is portrayed as a mindless, reanimated corpse with a hunger for brains. Thus Frankenstein's monster is not considered a zombie. Additionally, he was not brought to life through magical means or virus, but through science.
Both creatures are mindless, but Frankenstein's monster has a potential to learn and gain intelligence to some extend. He learns how to dress himself not long after his creation and learns to speak the French and German languages fluently by the time he is eleven months old. He is surprisingly spiritual and it was implied in the book that he was a vegetarian, preferring things like berries and nuts to meat.
Additionally, while zombies are soulless, Frankenstein's creature draws upon multiple souls of its many body parts. He is a timid, emotional creature who is only driven to commit acts of violence out of despair and anger. All he wanted in life was to share his existence with another like himself, to find a sense of belonging in a world filled with people that feared and hated him for being different from them.