A $1 note and four quarters are both real money, but how (and when) they become real money is different.
How the Fed accounts for bills
The important thing is that the Fed's stockpile of cash that it gets from the Bureau of Engraving and Printing isn't "real money" (i.e. M0) yet. That cash is only used for fulfilling withdrawals from reserve accounts. You can't use it to buy a car, or a house, or to pay Janet Yellen's salary...no one, not even Janet, can use even a single dollar to buy a Diet Coke. In other words, it's not really an asset. Or a liability. Or anything but a stack of paper in the Fed's vaults that looks astonishingly like money, but isn't.
The upside to this is that when the BEP makes a delivery of fresh bills to the Fed, or when the Fed destroys old, ratty bills that aren't usable anymore, the Fed's balance sheet doesn't change. That's good! Those are practical considerations, not financial ones.
The downside is that there's nothing on the asset side of the sheet to explain how the Fed's liability to a bank is reduced when the bank makes a cash withdrawal from its reserve account. So the Fed balances it's balance sheet by recording an increase in a different liability: the cash in circulation. Kinda like transferring the balance on a credit card: you're paying down one liability by increasing another one. It looks a bit silly, but less silly than recording the destruction of old bills as an "expense" of the face value of the bills.
How coins are different
Coins, on the other hand, become money as soon as the Treasury gets them from the Mint. If they wanted to, they could pay their employees' salaries by ordering coins from the Mint for cheap, giving them to their employees at face value, and reaping fat profits as a result. In fact, that's pretty much exactly what they do: when someone wants quarters, the Treasury mints them at less than face value and then sells them at face value.
In practice the Fed does all of this buying and then distributes the coins according to demand. The important thing is that this bunch of coins is not like the Fed's stockpile of bills. It's already real money, and therefore shows up as an asset.
How much does this distinction matter?
Not much. If the Fed bought coins at the cheaper price from the Mint instead, stockpiled them like they stockpile bills, then distributed them as usual, the extra profit just goes to the Treasury anyway, like all the Fed's profit does. However, as things are, the Fed's purchases of coins are recorded on the Fed's balance sheet at face value, which is kinda silly.