"Abolition" is a long standing word that has been recognised and understood for generations.
"Abolishment" has recently gained popularity as an alternative. Grammatically, it is acceptable but it is a change that grates on the ear and adds nothing to the established use of English.
It is interesting that it is most widely used amongst those for whom correct English is, generally, something of a challenge and it is often used by those who do not know commonly used words and phrases, or who use them incorrectly.
Therefore, the use of "abolishment," while technically not wrong, marks the user out as not being fully competent in English.