What’s the difference between inflation and inflation tax?

It IS going to the government. What causes inflation is when the government increase the money supply faster than the economy is growing.

Imagine a simpler monetary system with no fractional reserve banking and no electronic money. All money is paper bills. The government prints the paper bills, and then uses this paper money to buy things, thus inserting the new paper money into the economy.

So suppose that the economy of a certain place at a certain time is producing, say, 100 million unit of "stuff". There is 100 million dollars -- let's call the currency "dollars" for convenience -- in circulation. So each unit of stuff must cost $1.

Then the economy does not grow, but the government prints another $10 million of currency and uses it to buy things. What happens? There's still just 100 million units of stuff being produced. So there's now $110 million to buy 100 million unit of stuff. The price of each unit of stuff must rise to $1.10.

The government used this $10 million it printed to buy stuff. At the higher inflated prices, $10 million buys about 9 million unit of stuff. That 9 million unit of stuff would otherwise have been bought by private parties. So the government has taken 9 million units of stuff out of the 100 million, leaving only 91 million for private citizens. The government effectively taxed the people 9%. But the tax was subtle, because they didn't actually hand any currency over to the government. They paid the tax in the form of lower value for their money, i.e. an "inflation tax".

This isn't a hypothetical example. Many governments have tried to finance government programs by printing money. The result is pretty much always inflation.

Real life is more complex than my simple example. Most money today is not paper and coins but electronic entries on a computer. An increase in the money supply of X% does not necessarily result in EXACTLY X% inflation, because there's the issue of "velocity of money, that is, how many times is each dollar spent in one year. Etc. But the principle is the same.

I believe the best way to address monetary inflation by the government (a.k.a. money printing) is by referring to it as a "hidden tax." Taking the US as an example, over the past 100 years, prices in the US have roughly increased by 5% per annum, and the purchasing power of the dollar has also dropped by about the same amount. A fancy way of describing an increase in the US monetary base is the Fed "expanding its balance sheet."

Money printing via fractional reserve banking is an excellent way for a government to reduce its debt burden. Due to something called the Cantillon Effect, when the Fed (or any other central bank) creates new currency, it gets to enjoy the full purchasing power of that newly created currency. This currency can be used for all sorts of things, not the least of which are paying off debt obligations. Governments love the inflation tax, because they don't have to explicitly ask citizens to pay it. Instead, it happens under the covers, in stealth mode. The problems arise, again due to Cantillon Effect, when the newly printed money works its way through circulation. As that happens, prices of everything will tend to rise. By the time the printed money reaches the average person, it buys less than the equivalent amount of currency held in the hands of citizens purchased, before the central bank printed that money.

Rest assured, central banks around the world, in particular the Fed and ECB, have every intention to try to print their way out of their current debt conundrums. Only time will tell if they can do this without a loss of confidence in their currencies taking place.

The "tax" in "inflation tax" is metaphorical. It means that although you have, say $100, in your bank account, your actual purchasing power is less than you would expect because of inflation. From your point of view it's as if someone came along and took a little bit of your money, like the tax man does.

It's actually quite common to use "tax" in this metaphorical way, even when the government isn't involved, and even when the resource being "taxed" isn't money. For example, if your coworker Steve is overly chatty, then you might be reluctant to ask him a short question because of the "Steve tax" of having to listen to him talk about his favorite sports team for 5 minutes before you can get out of his office. See https://en.wiktionary.org/wiki/tax, sense 2, for another example.

Depending on the situation, inflation can actually result in a transfer of wealth to the government.

Inflation is bad for creditors and good for debtors, because while any monetary debts stay the same nominal value, the purchase power those debts represent decreases.

So when a government has a high amount of debt, then intentionally causing an inflation can be a way to reduce the purchase power of that debt. Which means that there is a net transfer of wealth to the government.

This is most obvious with those citizens who invested into government bonds, and will now receive a coupon which represents less purchase power than they invested.

But it is also indirectly the case with citizens who have their money in a regular bank account. Sure, you might think you are losing money to the bank, because they now owe you less wealth if you decide to withdraw your money. But where does the bank gets its money from? The central bank, which is run by the government. So indirectly you are losing that wealth to the government.