Hi Luiz,
The Closest Facility and OD Cost Matrix are very similar solvers that often do very similar things, but there are some nuances to each which make them better for certain situations.
The Origin Destination Cost Matrix (ODCM) only solves in one direction, from the origins to the destinations. The Closest Facility (CF) solver can solve from facility to incident or from incident to facility.
CF can return output shapes as the true shape of the route, with or without measure values, straight lines or no output shape at all and it can also return directions for each route. The ODCM can only return either straight lines or no shape at all and it will not return any directions. Because the ODCM does not return complicated geometry or directions, it finishes more quickly than the CF and is meant as a faster alternative to CF when dealing with large solves.
There are some other differences, both on the front end and back end, but those are the main ones. If you're interested in learning more about the algorithms for each analysis, check out Algorithms used by the ArcGIS Network Analyst extension—Help | ArcGIS Desktop .
Hope that helps!
-Rachel
