In the case of the U.S., the deflation trauma of the great depression has not been forgotten. It is further kept alive by a Fed chairman who is an avid expert on the topic. It therefore should come as no surprise that in the current economic turmoil, the Fed would have a loosening bias for its interest rate policy.
The ECB, in contrast, has embarked on a tightening bias, citing the growing risk of inflation. Part of the decision is due to its single mandate of price stability (versus the Fed's dual mandate of price stability and employment). History is the other major factor influencing the decision making process of the ECB. More specifically originating from its dominant German Bundesbank heritage. During the interwar years, Germany had suffered a severe bout of hyperinflation. Other European countries were to follow suit during and after the Second World War. We can therefore conclude that the hyperinflation trauma was collectively experienced in Europe.
These differences (historical and political) explain to a large extent why the two central banks, although facing similar issues, have opted to tackle it in opposing ways. What is certain is that both cannot be right and it is unfortunately too early to say who is using the correct antidote.