In my opinion "spending vs. savings" is not the best way to put it. It would be better phrased as "debt vs. savings", or maybe even "growth vs. stability". I don't think even an Austrian School economist would argue that spending doesn't drive an economy -- that's what an economy is -- but rather that spending driven by debt is problematic. The argument is that more debt in an economy creates inflation by definition and destabilizes the market. They will cite countries that have less debt (among businesses and individuals) tend to have more stable economies and higher exports. Of course, this is partially because they are selling to countries with higher debt, so I don't think it's as simple as either.

EDIT: Also important to point out that Austrian school economics are against manipulation of monetary policy. They argue that interest rates, credit and savings ought to be driven by market factors rather than the government. It's not so much "debt is bad", but rather "artificially driving up debt by manipulating monetary policy is bad".

It's more like investment vs. consumption.

Keynesians would say the government should employ policies to increase overall consumption in the economy (aggregate demand).

Austians would say government shouldn't do either, but rather allow the free market to find the correct balance between consumption and investment. They are particularly against monetary policy because it increases overall investment in an economy (aggregate supply). The problem is that artificially high rates of investment inevitably lead to businesses offering goods and services that cannot be sold profitably (malinvestment).

That's why Austrians believe that artificially low interest rates cause business cycles.

> If savings are put into banks or investments, is there even a significant difference between saving and spending (for the whole economy)?

There's a difference in the velocity of money in the domestic economy; it’s not fundamentally tied to savings vs. spending, but is correlated to it.

The idea that banks lend savings is wrong.

A bank is not constrained by savings to lean. When a bank lean money is creating that money as a liability for the borrower and a asset for itself.