A pang is a serious, stabbing pain.
It may share an origin with prong, as in the prong of a pitchfork (which could definitely cause some pain!). A pang can refer to a painful memory, or a sudden feeling of regret, but most often it's used to describe the discomfort of hunger. Hunger pangs are stomach spasms, urgent signals from your gut begging your brain to go find food.
Here are some examples to chew on:
If you do feel a few hunger pangs, you may need a light snack. (Seattle Times)
Walking for five minutes every hour during the workday could lift your mood, improve your concentration levels, and even dull hunger pangs, research suggests. (The Guardian)
A pang is a kind of pain, to be sure, but the phrase hunger pains is just a mishearing of hunger pangs. While it's not grammatically incorrect, no linguistic or semantic use is served by swapping the more specific word, pangs, for the more generic word, pains. Pang already connotes "sharp pain," while pain would need to be paired with an adjective like sharp, acute, or keen to express the same sense. The beauty of the word pang is that the acute, piercing nature of the pain is built right in.
The examples below would both benefit from the specificity of the correct word:
The wheat ships were still not full enough nor fast enough to assuage the world's hunger pains. (Time Magazine Archive)
But then he thought of Trina, and at the same time his hunger pains roared in his belly. (The Kill Order)
English is blessed with many different words for similar things. Part of the fun in mastering the language comes from understanding the specific uses for certain words. You could say "Hunger caused sharp, insistent pain in her stomach" — or you could simply say "She felt hunger pangs."