Zeus and Jupiter are the most well known gods from Ancient Greece and Ancient Rome. Zeus was the king of Olympus (the mythical area where the gods lived in Ancient Greek mythology). His area of control over the human population was the Sky and his symbol was a mighty golden thunderbolt. Jupiter was the leader and ruler of all gods and man in Ancient Rome (on a timeline, coming after ancient Greece). He too was lord of the sky and his symbol was also a powerful and mighty thunderbolt.
The similarities between these two gods of two ancient civilizations however end there. Zeus was a supreme God; however had several human attributes such as emotions of love, jealousy, and contempt. He was seen as frivolous, and often portrayed as careless and easily influenced, particularly by female gods, who would use their charms on him.
Jupiter however, in Ancient Rome, was portrayed as a stoic leader, lacking emotion entirely (like most Gods in Ancient Rome) and his method of ruling was often compared to a board-room- organized, with a few advisors; however, the ultimate decision always lay with Jupiter. While Zeus was seen as fickle and careless, Jupiter was portrayed as calculated and reason-driven.
Primarily, Zeus and Jupiter are the same god, controlling the same realm, just through two different civilizations. The Ancient Greeks existed before the Romans, and so it can be argued that Jupiter is a recantation of Zeus, with subtle changes that reflect the changes taking place in society. While the Greeks viewed Gods as humans, bestowed with special powers and immortality, the Romans saw their gods as moral bastions and unattainable ideal forms. As such, during the time of the Greeks, myths of the gods involved errors in judgment (as humans make) and attributes of jealousy and revenge. However for the Romans, Gods were perfect, unlikely to make mistakes and thoroughly well reasoned.