In simple terms, think of "income" as what you get paid (and I'd include government handouts in this, as you indicated.
Think of "wealth" as what you have left over after you pay your bills.
In any society, there is pretty much a direct correlation between how much money you actually earn (as opposed to getting handed to you) and how much wealth you produce.
Yes, because what you earned is going to be precious to you. What someone hands you for doing nothing will be frittered away, because it didn’t cost you anything to produce. It destroys a person’s dignity and sense of purpose to be kept like an expensive pet.