I'll state from the get go I am new to Alfresco and a search has not returned a definitive answer. I did RTFM, but could not find a comparison or reasoning.
As I understand the WAR is a standard Java app and thus your installs Tomcat, database, and others can be maintain separately. If you use the binary installer then those systems are maintained by the Alfresco crew.
There's no "best way". Both methods are equivalent.
However novices should start with the binary installer which will get them up and running quickly without a learning curve. Experienced users may well do a manual install of the WAR since its quicker and configures exactly as you want it.