Formally, if $f$ is a function, $f(x)$ is a value. So for instance, $f$ can be continuous, but not $f(x)$. 
In teaching at school and university, notation is quite often mixed up, e.g. the function is called $f(x)$. This may be problematic but can also have benefits. **What good reasons for the use of the different notations do you know?**

This question raised from the discussion of [this discussion on notation of derivatives](https://matheducators.stackexchange.com/questions/562/should-students-be-asked-to-use-more-than-one-notation-for-the-derivative-in-an).