Modernity is a historical period in the Western world beginning roughly at the Renaissance. The term generally refers to societal movement away from the traditional Christian metaphysic that emphasized received authority and moral virtue, and towards the 'modern' metaphysic which emphasizes reason, technology, and liberalism. The modern era celebrates the greatness of man and mankind, where the earlier era subjugated man and mankind to the power of divinity.
Modernism is a school of thought that grew through the Colonial era, in which Western ('modern') civilization was held as the epitome of cultural development, superior to all other forms of civilization. Modernism was largely a Protestant movement (Catholicism was largely seen as a holdover from the pre-modern, less-civilized world) that imposed Western commercial systems, Protestant ethics, science, and technology on other regions. In some cases it imposed these values reflexively, as part of the ongoing capitalist expansions that ran through the Colonial era; in others it consciously imposed these values with the 'White Man's Burden' intent of lifting other civilizations out of ignorance.
In the early to mid 20th century there were a set of philosophical and cultural movements that questions and/or opposed the blithe assertion of Western cultural superiority. People with a modernist mindset often lump these various movements under the rubric 'post-modernism', though that terminology is not generally used by the movements themselves. The essential idea of all these movements — in one sense or another — is that modernism confuses technological superiority with cultural virtue (the 'might makes right' problematic), and that 'Western culture' is not superior or inferior to other cultures, is not everywhere morally coherent and consistent, and is not even (in any real sense) a singular or well defined thing.
I'm not sure who calls neoliberalism post-modernist. Neoliberalism (as best I can tell) actually calls for a return to early (18th and 19th century) political philosophy. They are still purely modernist, with perhaps the tweak that they do not believe the modernist programme has completed itself, and seek to undo certain socialist and humanistic reforms that undermine the modernist agenda.