The Dynamics 365 Web API did not exist in CRM 2013, so you cannot use that for both purposes.
Instead you can use the Organization Service (also called the CRM 2011 endpoint). The behavior of this endpoint has been relatively consistent (with some minor changes to how e.g. updates should be made) from CRM 2013 to D365.
Be aware that the Organization Service will be removed at some point in the future, but currently works just fine:
The Web API is intended to replace the CRM 2011 endpoint, also known
as the SOAP endpoint, for accessing the Dynamics 365 organization web
service and other provided web services.
As of the release of Microsoft Dynamics CRM Online 2016 Update 1 and
Microsoft Dynamics CRM 2016 Service Pack 1 (8.1.0), the CRM 2011
endpoint has been deprecated. The 2011 endpoint will be removed some
time after the release of Dynamics 365 version 9.