Ascent means (1) the act of rising or moving upward or (2) an upward slope. It’s the opposite of descent. Assent means agreement or acquiescence. Assent may also be used as a verb meaning to agree, while the verb form of ascent is ascend.
Examples
Here are a few examples of ascent and assent used correctly:
The speaker praised BP’s decision to assent to demands that they create an account in escrow to pay out claims it faces as a result of the massive oil spill in the Gulf of Mexico. [The Hill]
The Benders have made a dizzying ascent into indie stardom since their exodus East. [San Francisco Chronicle]
New laws only come into force once they have been assented to by the president and published in the Government Gazette. [Independent Online]