According to wikipedia economic statistics would also include the analysis of the economic data. I think it's a philosophical issue and I would consider both of them as synonims in a wide range of contexts. Anyhow, in some specific situations I would take the distinction made by Balazs as an excellent definition.
1. Economic statistics is collection of data and analysis thereof to arrive at any definite conclusion, if at all, regarding states of an economy. On the other hand, econometrics is also collection of data, but the collection of data is towards empirical testing of an established theory.
2. Economic statistics measures economic variables, not necessarily keeping in view an established relationships between them, whereas econometrics measures variables, keeping in view an established relationship between them.
3. Economic statistics deals with economic variables but econometrics deals with any variables (not necessarily economic in nature).