I wonder if there is any difference in using approve (of), assent, consent, and grant as verbs. Can anyone help?
"Consent" and "assent" are almost synonymous, and mean "allow" or "give permission for". They generally have to take a noun object, not a clause: "They consented to his doing it", not *"They consented that he could do it".
"Assent" is perhaps more formal in this sense; but in literary use "assent" can also mean "say yes", without necessarily implying permission.
"Grant" also gives permission, but you have to say what permission, or access, or rights are being given. You can't say *"They granted him to do it" or *"They granted him that he could to it", it has to be "They granted him permission to do it" or the like.
"Approve" has two different meanings. Without "of", it means "give permission for": "They approved his plan", "They approved his going ahead with it". It can't usually take a clausal object, so not *"They approved that he did it".
"Approve of" means "like" or "agree with", but has no implication of giving permission, or of the authority to give permission.
As Google ngram shows (looking for past participles because some of these verbs can also be nouns), assent is the least used of the bunch, with grant and approve leading the pack.