The Gupta Empire ruled much of north India from about 320-550 CE and the Mauryan Empire ruled much of the same territory of north India from 321-185 BCE. Among their similarities:
1. Both ruled over large areas of north India, and both expanded their territory - basically became empires - through conquest.
2. Both were periods of advancement in various areas. For example, under the Mauryas, the social and political manual "Arthashastra" was written to explain statecraft. The Gupta Empire saw the development of mathematics and astronomy and literature (such as the plays of Kalidasa).
For some of their differences:
1. the Mauryan emperors supported different religions: Chandragupta Maurya followed Jainism, helping it rise to prominence, and Ashoka was a follower of Buddhism, and sent Buddhist missionaries abroad (incidentally, the Buddhist message of nonviolence he promoted after expanding the empire in conquest). On the other hand, while there were significant Jain and Buddhist populations in the Gupta empire and the empire promoted rleigious tolerance, the rulers were Hindu.
2. The Mauryan Empre ended through internal strife: assassination of a ruler and a coup brought down the empire. The Gupta Empire faced external threats with military conflict against the Huna people of northwest India/central Asia.