Why is there a 1 cent piece and a 1 dollar bill, 5 cent and 5 dollar, 10 cent and 10 dollar, 50 cent and 50 dollar, 1 dollar coin and 100 dollar bill but…25 cent piece and 20 dollar bill? I guess a “quarter” sounds better then a “fifth”.
Waal…there used to be a twenty-cent piece, too (along with the quarter) but it was unpopular because the two coins tended to be confused. I guess it’s usual to divide things (such as a dollar) into halves, and then into halves again, and there’s your quarter. A hundred dollars is just too high an amount (too high, especially, in centuries past) to think about dividing it like that.
Well… my 25 cents…
Coins are a pain to carry, so the quarter allows for less coinage in MOST situations (personally, I throw pennies away).
OTOH, people like to have big wads of cash, so a $20 bill allows more bills to be carried, thus a bigger wad.
Honestly though, I would think there was no reason at all for it, they just said “yeah… $20 bill… I like that”.
On another note:
Was “grabbenmanuts” already taken when you picked your username? 
That’s kind of a lame answer. Also, I thought that the discontinued coin in question was a bit, which was actually half a quarter or 12.5 cents. (You know, shave and a haircut…) Other than making it somewhat easier to make change for a buck, anyone have an official answer as to why the quarter, and why the $20?
I don’t know about the quarter, but it seems to me that a 20 is easier to count then a 25 in situations where you were counting bills quickly.
Yeah, there was a “bit” – they used to take the Spanish milled dollar (or real) and literally cut it to bits. But after that, in the nineteenth century, there was an official U.S. 20-cent silver coin, very like the quarter.
I still think my answer’s pretty good: a hundred dollars has not traditionally been thought of as a unit to be divided (half, then half again = quarter); rather, it’s the dollar that’s been considered as a unit to be multiplied (decimally, twice ten = twenty).
scratch1300 is probably right as to the basic reasoning, but there are other factors at work as well.
There are many US coin denominations which are no longer minted. As well as a variety of sizes and designs for the same denomination. Any government which has been issuing money for a long time will have the same situation. Look over the welter of coinage issued historically in the UK.
The basic answer is that some prove to be useful and/or popular, some don’t, for a variety of reasons. Half-cents, two cent pieces, three cent pieces and twenty cent pieces have all been issued by the US mint, as well as various denominations in gold. Some of the reasons for issuing them can be pretty capricious. The silver three cent piece (the smallest US coin ever issued, in physical size) was created because stamps were 3 cents at the time, and it would be a convenient coin to use at the post office.
As someone pointed out, people reacted to the twenty cent pieces about the way they reacted later to the “Suzie B”. They just didn’t accept them, and found them too easy to confuse with the quarter.