Abolishment appears in many dictionaries and is not considered incorrect, but abolition is preferred in all modern varieties of English. Both nouns mean the act of doing away with something, and neither has any meanings it does not share with the other.
Both words date from the early 16th century (soon after abolish came to English from French roots), but abolition has always been more common, and it now appears about ten times as often as abolishment. Some writers reserve abolishment for senses unrelated to slavery, but the distinction is unnecessary.
In these examples, abolishment is not wrong, but the more common abolition would work just as well:
Top EU officials, while speaking in favor of the measure, have said that full abolishment of visas won’t happen in the near future. [CNN International]
Among the topics broached in this effort was the removal of the two-line pass and the abolishment of the trapezoid area behind the net. [Fox News]
And these examples show that abolition can denote the doing away of anything (not just slavery):
Within hours of the first reports of trouble at Japan’s nuclear power plants, calls for abolition could be heard around the world. [Vancouver Sun (article now offline)]
The abolition of prescription charges in Scotland has sparked division and anger … [Scotsman]
Supporters of negative gearing argue that its abolition would lead to a ”landlords’ strike” … [Sydney Morning Herald]