The extra cent goes to brain tax. It is actually a reminder. Being *asked* for your opinion is a lot more valuable then just spouting off your own opinion, because if you are asked then you get a penny but if you just spout off then you have to give them 2 cents. (Yeah, I'm an accounting major, lol)
Really though if you want the nerdy homeschooler answer, the sayings probably actually have two completely different origins. I am guessing that "putting my two cents in" had something to do with poker and anteing up. But I have no clue about the penny for thoughts one, and I am actually just making this all up.
As far as pennies vs cents. A penny is 1 cent. A dime is 10 cents. But a dime is not 10 pennies. You can have 10 pennies and it will make 10 cents, but it won't make a dime. A penny is the name of a denomination of money, a cent is the actual money. Like a quarter is 25 cents. But if you have 25 pennies you wouldn't say you had a quarter and if you had two dimes and a nickle you wouldn't say you had a quarter, but you would say you had 25 cents. Ok, so you are more confused now. Oh well.