The difference is that you can pay someone with JPY and that's the whole transaction. When you pay someone in BTC, you also have to pay the power company with something. The system is an inherently leaky bucket.
The "fee" for fiat is inflation. Your money is regularly losing value even if you do not transact it. Not to mention cash transactions as you describe are on the decline and do not represent the majority of transactions that happen in the U.S.
I'm not saying this is a bad thing by any means, in some ways it is a good thing. Just stating that it exists.
Does cash just grow on trees? No, it is printed by a government that employs thousands of agents who have to make sure that no one is counterfeiting those bills. The fact that I can hand someone a $20 and walk away with my bag of groceries is the tip of a massive iceberg of regulatory and enforcement mechanisms.
