The OED, in the definition of syllepsis reports that's another term for zeugma.
The Collin English Dictionary reports syllepsis means:
(in grammar or rhetoric) the use of a single sentence construction in which a verb, adjective, etc is made to cover two syntactical functions, as the verb form have in "she and they have promised to come."
Also the Collin English Dictionary reports that syllepsis is another word for zeugma (as second meaning).
Yes they are different, although the definition and distinction my only be of interest to scholars of classical rhetoric. There are, according to this article from the University of Alberta, three figures: Figures of Word, Tropes (Figures of Speech), and Figures of Thought. All three are tools in rhetoric for conveying information.
Figures of word involve the actual construction of a sentence, including word selection, placement, and repetition.
Figures of speech are when words or phrases are used in a sense other than their literal meaning for dramatic effect. Hyperbole is a figure of speech.
Figures of thought are the styles that rhetoric can take, the way an argument is approached. A simile is a figure of thought.
They are different concepts, but not mutually exclusive. Examples of figures of thought could be figures of speech, and contain figures of word. Figure of speech is, by a wide margin, the concept that we are most familiar with. And we are familiar with specific types of all of them, including, hyperbole, paralipsis, metaphor, and allegory, but their technical classification wouldn't be of much interest to anyone outside of an academic setting.
Best Answer
"Of the people" refers to the government's composition. It's referring to the fact that the government is made up of people who come from the people.
"By the people" refers to who chooses those people who make up the government.
Essentially, it's saying that it's a government comprised of common people who were chosen by common people.