How do this scenario differ from an economy experiencing deflation?
One of the differences is, that overall the money is still losing value - you only make up for it by the interest the bank is willing to pay you. The interest rate a bank pays is usually bound to a base rate (federal funds rate) controlled by the government. The government therefore has still control over the development of the monetary value and it will usually avoid raising the base rate to a level which would equal significant deflation.
Inflation can be limited by increasing the base rate, therefore giving people more incentive to save instead of invest. This can be used to slow down economic growth which might otherwise be too fast (as currently in China for example).
Deflation on the other hand cannot be limited indefinitely via governmental control of the base rate, because if the base rate is already zero it can't be lowered any more.
Therefore, in a deflationary economy, the government usually has little control over the development of the economy. If it is already shrinking and money is deflationary (for whatever reason) it might cause the infamous deflationary spiral and it is very hard to get out of it.
Control of the development of monetary value (degree of inflation / deflation) via the base rate is probably the most direct and effective way a government (or rather the Fed) can influence the growth of the economy - unfortunately this control mechanism only really works for inflation.