Abolition vs. Abolishment

It is most likely that "abolition" is the more common form due in large part to its association with the "abolition movement".

France was one of the earlier countries to abolish slavery within its borders, and Société des amis des Noirs was one active group in the movement in France.

The picture on that web page refers to (pardon my French) L'Assemblée Nationale pour L'Abolition de la Traite des Noirs and is dated 1790. Clearly "abolition" was the preferred term in France, and that preference would have transferred to England.

Further, those advocating abolition in England and the US were "abolitionists". It would have been a mouthfull to call them "abolishmentists".

The main reason one might choose "abolishment" over "abolition" (aside from the subtle difference in meaning) is that "abolition" is associated with the movement to end slavery, and thus might be confusing or distracting when used to refer to a different process. But this confusion is not likely in native English speakers, unless the context contributes to the confusion.

According to The New Fowler Modern English Usage -

"abolishment and abolition" both entered the language in the 16c. For two centuries they coexisted as synonyms (used of sins, faith, guilt, and other abstract concepts) with approximately equal currency.

The choice of abolition by those working for the abandonment, first of slavery, and then of capital punishment, ensured the relative supremacy of this member of the pair.

According to French Anti-Slavery: The Movement for the Abolition of Slavery in France, 1802–1848, by Lawrence C. Jennings,

The first French abolitionist movement centered around the Société des Amis des Noirs. Founded in early 1788 by the journalist Jacques Pierre Brissot and his associate Etienne Clavière, with the close collaboration of Count Honoré de Mirabeau, the Amis des Noirs was inspired by the humanitarianism and egalitarian currents of thought implicit in the eighteenth-century Enlightenment. It was also profoundly influenced from the moment of its inception by the British precedent. The Amis des Noirs was modeled upon the London Society for the Abolition of the Slave Trade, which had been formed in May 1787.

(p. 1)

Contextual needs, however, still leave room for abolishment to be used with fair frequency.

• He rewrote things to reflect this, now calling for the "abolishment of politics".

  ("Sex-Pol: Essays, 1929-1934 by Wilhelm Reich – review", Christopher Turner, The Guardian, May 1, 2013.)

• It's like calling for the abolishment of pedestrian crossings because some people skip red lights.

  ("Theatre bloggers must leave previews alone", Matt Trueman, The Guardian, Feb 10, 2011.)

• But its abolishment “doesn't remove moral and ethical responsibility,” they said in a joint statement.

  ("Adultery Is No Longer an Affair of the State in South Korea", Choe Sang-Hun, The New York Times, Feb 26, 2015.)

• As a nation, abolishment of slavery was and is an important milestone in our history.

  ("Ramadan, Day 12: Sex Slavery and Objectification of Women", Sohaib N. Sultan, Time, July 9, 2014.)