Nothingness is more of abstract, literary term, something a philosopher or a poet would be more prone to use than a scientist. It's a concept of an entity composed of nothing. The sentence doesn't speak about thermodynamic death, where everything comes to a standstill with no active energy sources left. It speaks about unending expansion that will spread matter so thinly over so big area there will be nothing (no meaningful amounts of it) left. It will be a universe infinitely bigger than current one, with the same, finite amount of matter filling it - spread over infinite distance, so that the vast volume is empty - it's nothingness.
Nothing is a vastly more utilitarian, day-to-day word. It's a joker card for lack of meaningful content. There's nothing in the box - not true, there's air but nobody cares. You worry over nothing - the subject of your worry is unimportant. "Nothing" can be quite big and important but it's beyond our interest.
Emptiness is the content of a volume, or a state of something. While nothingness is an entity, emptiness is its content. It's also a rather philosophical word, and rarely used in its practical meaning - degree of being empty (you don't measure emptiness of a container but its fullness). Much more often you'll find it in its figurative meaning - emptiness of stare, of expression, of thought or quotation (devoid of meaning or importance).
Void is a solid scientific word describing the area between concentration of matter - whatever in the universe isn't stars, planets, meteors, and so on. It isn't entirely empty, and has different properties in different areas depending on the trace amounts of matter it contains.
Void also has a common meaning, which is "unnatural lack of something which would normally be there". An emotional void is lack of normal emotions. A void in the market will soon be filled by entrepreneurs.
Vanity is definitely a figurative emptiness - a property of character, being extremely proud but of no real value as a person, lack of virtues.
Zero is a mathematical term, a value describing an amount. Air pressure in void is zero. The count of particles in perfect void is zero. The value of that vain jock in my eyes is zero.
There's also vacuum. Volume with extremely little matter in it. Perfect vacuum does not exist - there will always be some energy, some particles manifesting themselves spontaneously from quantum uncertainty, but generally lack of matter, including air is considered vacuum. It's often used interchangeably with "void", but you won't say that collapsing cistern was filled with void (or "just empty"), there was vacuum in it.
