Neoclassicism is, like the classical poetry it tried to imitate, characterized by objectivity, rationalism, and realism. This is in contrast to the much more emotional emphasis of Romantic poetry. Neoclassical poetry is also often didactic, meaning that it tries to educate the reader. Stylistically, neoclassical poetry is characterized by heroic couplets; historical, religious, and biblical allusions; and restrained, concrete language.
In Samuel Johnson's "The Vanity of Human Wishes," there are several academic allusions, whether it be historical ("Democritus, arise on Earth . . . In full-blown Dignity, see Wolsey stand, / Law in His Voice and Fortune in his Hand"), mythological ("Though dancing Mountains witness'd Orpheus near") or literary ("The Tenth Satire of Juvenal"). These allusions are to lend the poem weight and gravitas.
Johnson's poem is also didactic in that it attempts to draw attention to the ignorance and fickleness of the British people who:
ask no Questions but the Price of Votes;
With Weekly Libels and Septennial Ale,
Their Wish is full to riot and to rail.
Johnson's message here is that the British people should be more inquisitive and, by implication, less accepting of the corruption of their rulers, who, Johnson implies, cynically purchase their votes. He also criticizes the British public for wanting nothing more than to satisfy base instincts by drinking and rioting. Later in the poem there is more didacticism as Johnson rails against avarice, describing dismissively the greedy man who "Unlocks his Gold and counts it till he dies."
The poem is also written (in accordance with the neoclassical style) in heroic couplets, which are pairs of rhyming lines written in iambic pentameter. An iamb is a metrical foot of poetry, in which a stressed syllable follows an unstressed syllable. Pent (from the Greek "penta") means five, so there are five iambs in each line. For example:
Let Ob/servat/ion with/ exten/sive View,
Survey/ Mankind,/ from Chi/na to/ Peru
The syllables I have highlighted in bold are the syllables which are stressed, and you can see that the syllables are in pairs, or iambs.