What is the difference between how about ~ and what about ~ ?Feel free to just provide example sentences.

I don't think there is much difference. You can use them interchangeably except for one situation:

When you add a factor to something:
situation 1
A: "Tommy and Melanie and Patrick are coming tomorrow...uh...who else..."
(トミーくんとメラニーとパトリックが明日行く…えっと…もう一人がいた気がする…)
B: "What about Mark?"
A: "Oh yeah! Mark!"

-> How about Mark?❌

situation 2
A: "I'm gonna go see Grampa on Monday so I don't have to go to school!"
(月曜日にじーちゃんと会いに行くから学校に行かないで済むよ！！)
B: "...but what about the test on Monday?"

-> How about the test on Monday?❌

But if you are proposing a plan, then both are okay.
"How about we visit the aquarium?"✅
"What about we visit the aquarium?"✅
(水族館に行ったらいかが？)