Interesting question. Estimators and statistics do not need to be different things, though. They are different concepts.
A statistic is a function (in broad terms) in which the input is (statistical) data. The effect is that you gain a result, usually a number, from this statistic. In a more abstract term, a statistic may yield more than one number.
The statistic depends on the data, but the procedure is deterministic. So the statistic may be: "Sum all numbers and divide by the count" or, in the broader sense "take the gdp data and prepare a report on it".
In the statistical sense we are of course talking about a mathematical function as a statistic.
The significance of this is that if you know properties of the data you input (for example it beeing a random variable), then you can calculate the properties of your statistic, without actually putting in empirical data.
Estimators are estimators because of you intent: to estimate a property.
As it turns out, some statistics are good estimators.
For example if you pull data points out of a pool of i.i.d. variables, then the arithmetic mean - a statistic based on the data you pull, will probably be a good estimator for the expected value of that distribution. But then again any thing that produces an estimate is an estimator.
In practice, estimators that you use will be statistics, but there are statistics that aren't estimators. For example test-statistics - though one can argue about the semantics of this statement and to make matters worse, a test statistic may not only be but also include estimators. Though conceptually this doesn't have to be the case.
And of course you can have estimators that aren't statistics, though they are probably not very good at estimating.