【Advance】 , 【promote】 , 【forward】  and further all mean to move or put ahead, but they come into comparison chiefly when they imply help in moving or putting (something) ahead.

【Advance】 usually implies effective assistance, as in hastening a process.

• the warm rains greatly advanced the spring crops

Or in bringing about a desired end.

• the pact should 【advance】 peace among nations

Or in exalting or elevating a person, especially in rank or in power.

• Ahasuerus . . . advanced him . . . above all the princes
  —Esth 3:1

The implication of moving ahead is dominant in 【promote】 when the word means to 【advance】 in grade or rank, especially in a predetermined order.

• 【promote】 a pupil to the next grade in school.
• 【promote】 a member of a college faculty from associate professor to full professor

When the dominant implication is assistance, 【promote】 may suggest open backing or support.

• the objects for which a corporation is created are universally such as the government wishes to 【promote】
  —John Marshall

It may, especially when the subject names a person, his influence, or his acts, imply actual 【advance】 by encouraging or fostering.

• a sound forest economy promotes the prosperity of agriculture and rural life
  —G ustafson

It may, when said of a thing such as a practice, a policy, a habit, imply subservience to an end that may not be intended.

• the habit of regarding the language of poetry as something dissociated from personal emotion . . . was promoted by the writing of Greek and Latin verse in school
  —Babbitt

In one or two collocations 【forward】 implies not assistance but effective carrying out.

• 【forward】 a shipment by express
• please 【forward】 all letters during my absence

In its more common sense 【forward】 is often not clearly distinguishable from 【advance】 , except that it is seldom if ever used with reference to persons.

• Marie de Médicis had advanced Marillac by marrying him to one of her maids of honor . . . yet . . . she only forwarded the marriage because she wanted to do the girl a favor
  —Belloc

Further, less than any other word in this group, implies movement ahead and, perhaps more than any other, emphasizes the assistance given, especially in the removing of obstacles, either to a person in an undertaking or to the project he undertakes.

• her sole object. . . was to further him, not as an artist but as a popular success
  —Brooks
• bodies like the French Academy have such power for promoting it [genius], that the general 【advance】 of the human spirit is perhaps, on the whole, rather furthered than impeded by their existence
  — A mold