If you give your assent to something that has been proposed, or if a proposal has your assent, you agree to it.

1. Haldane acted in these talks with the full assent of Grey and the Foreign Office.
2.‘Cocoa all round?’ People murmured assent.
3. There is at least grudging assent to his rule.

If you give your consent to something, you give the permission that is needed for it to happen or be done.

4. I had to ring the education office to get official consent.
5. They ask for no-one’s consent to come, they just arrive.
6. Have you the husband’s consent to take this child out of the country?

You can also say that when a number of people all agree that something is the case, or that something should be done, they do it by consent.

7. The question of whether ordinary people can govern themselves by consent is still on trial.
8. By common consent they stopped.