Difference between Brushed ESC and Brushed DC Motor Driver (Controller)
The main difference is that ESC you linked is a complete module meant to be used by the end user. On the other hand, the motor controller you linked is just a bare component which is meant to be integrated in a module. So the target audience is different. Motor controller is expected to be used by an engineer (or at least experienced hobbyist) who has necessary electronics knowledge to make it work, while use of ESC often requires drastically lower level of electronics knowledge since often only thing that needs to be known is to follow assembly diagram. 

Motor controller is the main part of the ESC, but there (usually) are other components as well which are used on an ESC to provide complete module. For example, the ESC is supposed to be connected to a radio which will output some type of signal. That signal is not necessarily what motor controller expects, so there may be need for a conversion stage. 

Also ESC will often have a protection mechanism which will turn off motor when battery voltage is too low, while motor controllers often do not have that protection. 

Depending on the power ratings, ESC could use a motor controller made from discrete components instead of an IC motor controller.

The exact difference is a bit fuzzy, since there is a tendency of integrating as many features as possible on a single IC, so some features which ESC module would provide using components other than the motor controller would find themselves integrated onto the motor controller.