【Approve】 , 【endorse】 , 【sanction】 , 【accredit】  and certify all mean to have or to express a favorable opinion of.

【Approve】 often means no more than this.

• daring them . . . to 【approve】 her conduct
  —Conrad

Sometimes, however, it suggests esteem or admiration.

• Jane secretly approved his discernment
  —Rose Macaul&y

【Endorse】 adds to 【approve】 the implication of backing or supporting (as by an explicit statement): it is therefore used chiefly in reference to things requiring promotion or publicity.

• 【endorse】 a person’s candidacy
• 【endorse】 the platform of a new political party
• 【endorse】 a brand of cigarette

【Sanction】 not only implies approval but also authorization.

• the school dances were sanctioned by the board of education

The one that sanctions may be not only a person or group but something that provides a standard by which something can be approved and authorized or disapproved and discountenanced.

• proposed laws not sanctioned by public opinion
• some churches permit divorce, but do not 【sanction】 remarriage
• these statements are sanctioned by common sense
  — Joseph Gilbert
• the court has also sanctioned recently some federal efforts to protect Negroes in the South from violence
  —Barth

【Accredit】 and certify usually imply official endorsement and conformity with certain standards. Their selection is dependent on idiom rather than on distinctions in meaning.

• an accredited herd of dairy cattle
• certified milk
• an accredited school
• a certified teacher
• a certified public accountant
• labels by which brain merit is advertised and certified—medals, honors, degrees
  — Woolf

Accredited, however, is sometimes used generally as implying public approval or general acceptance.

• ifany . . . break away from accredited custom
  —Inge
• sages so fully accredited as Mr. Bertrand Russell
  —Montague