This is the question at the heart of all central banking orthodoxy in the late 20th century, and the answer is credibility in fighting inflation. Tony Blair made the Bank of England independent because this was the standard economic advice at the time, as I will explain below. Bill Clinton did not make the Federal Reserve independent, because it already was independent and had been since its creation with the Federal Reserve Act of 1913.

To explain in more detail: The central bank is the entity that regulates the money supply. (Exactly how their various mechanisms for doing so work, and why, is still a matter of some considerable dispute. But everyone agrees they at least try and do this.) Ever since the Great Depression and the widespread adoption of Keynesian insight that the government could use the money supply to offset booms and busts, the main role of central banks has been to maintain the stability of the price level. Typically, this involves setting an inflation target - 2% being the most common by far. (Sometimes it also includes an explicit full employment mandate, as with the Federal Reserve.) This means increasing the money supply in bad times, to prevent deflation and output loss, and "taking away the punch bowl" in good times to prevent overheating and asset bubbles.

So, if this is a useful and widely (if not universally) acknowledged role for central banks, why is it a problem if they are nationalized? Why is an independent central bank seen (by the late 20th century) as the fundamental anchor for a country's macroeconomy? The problem is incentives, and the public perception of those incentives.

Money has three elemental functions: it is a medium of exchange, a store of value, and a unit of account. Insofar as something fulfils those functions, it is "money-like." The value of money, like any asset, is tied to expectations about its future. If money buys a lot today, but you know it will buy very little tomorrow, then you have an incentive to buy non-money goods today. But that signals to the market that you are willing to part with your money - it becomes less valuable today, because you have increased the supply of it on the market. And if everyone takes their cues from your behaviour, then expectation of tomorrow's cheap money will be enough to crash its price today.

Why might one expect the price of money to drop? Supply and demand gives us two elemental possibilities: The demand for it is expected to go down, and/or the supply of it is expected to go up. One reason the supply of money might increase is that the government might want to spend more than it taxes. In that case, governments must either pay directly in money, or to issue bonds, which are effectively a promise to pay money in the future (plus some interest). If those bonds are denominated in the sovereign currency, then their worth is in part determined by the value of the currency - by the price level and exchange rate. (If the bonds are denominated in foreign currency, or gold/silver/whatever, then this is "original sin" and can lead to very serious macro problems if the exchange rate crashes.)

If the monetary authority is separate from the fiscal (taxing and spending) authority, and they are tasked with price stability, then their reaction to increased government deficits should be to conduct tight monetary policy, to "lean against" the inflationary effect of expansionary fiscal policy. This should raise interest rates, making government borrowing more expensive, and forcing the fiscal authority to borrow less/pay back more. (Or this can all happen the other way around, in theory, if the government is running large surpluses. Though this scenario does not seem to scare most central bankers in quite the same way as rampant deficits.) If the central bank is powerful enough to offset the fiscal effects, then we say there is "monetary dominance," and the macroeconomy dances to the tune of the central bank, not the fiscal authority.

But if the fiscal authority is strong enough to overcome the central bank's interventions, then we have "fiscal dominance," and the macroeconomy dances to the tune of government spending. This is kicked into overdrive if the monetary authority and fiscal authority are working together to finance government deficits with monetary expansion at cheap rates. This is a way to overcome fiscal limits, to increase the government's spending without the associated painful increases in taxation or cutbacks in spending elsewhere. Unsurprisingly, this is a favourite of politically-constrained governments everywhere. But it can have severe consequences - the most extreme of which is hyperinflation.

This causes peoples' expectations of the future money supply to increase: people expect inflation, that their money will be worth less in the future than today. Money can continue to function as a medium of exchange, because the present is instantaneous, but as a unit of account or a store of value, people will begin to adapt their expectations and either get rid of their money quickly, or write contracts (loans, mortgages, salaries, utility bills, etc) that adjust for this increase. (This is called indexing - clauses like cost of living increases.) Inflation is effectively a tax on holders of money, and people will adapt their behaviour to reduce their exposure to that tax. If the government then wants to keep using monetary expansion to finance spending, the nominal amounts they need will increase by an even faster rate, to outrun the indexing. This is how you get hyperinflation, which will not stop until people expect, for one reason or another, that monetary expansion will cease.

The incentive problem here is thought to come from government pressure on central banks to accommodate rather than counteract deficit spending. So long as a central bank takes its orders from the fiscal authority, there is the worry of fiscal dominance. And if the fiscal authority is ever less than credible at balancing its budgets over the long run, there will be inflation.

In order to mitigate this risk, states choose to voluntarily relinquish control of their central banks. This is a way of tying their own hands, to assure the public that monetary dominance will continue. Thus, savers and investors can feel confident in using money as a store of value and unit of account without fear of its value changing unpredictably. This is the core of central banking orthodoxy in the post-Gold Standard era, a way of reconciling the phenomenal power of modern sovereign money with the temptation to use that power for short-term purposes.

This conception of the risks of inflation and the role of central banks was heavily influenced by the inflations of the late 1960s-70s (and in Britain the trauma of the crisis and devaluation in 1967), the sour experience with more direct monetarist policy in the early 1980s and the ill-fated attempt to join the European Exchange Rate Mechanism in the early 1990s to control inflation. An independent central bank maintaining a floating sovereign currency and targeting a modest (2%-ish) inflation rate was the "modern" solution. Our thinking about central banks has changed at least somewhat since then, but the era of QE is outside the bounds of this subreddit, unless you're specifically interested in Japan rather than the UK.