I have written a simple Apex class that has @RestResource annotation set so that I can invoke methods in this class via REST from our own application. The Apex method I created does a simple insert into the Opportunity table based on parameters passed into it and this function seems to work fine (i.e. the Opportunity record is created as expected).
 
We also have a simple Process Builder workflow set up that performs some automated tasks when a new Opportunity record is created; in this case for example, it creates a new Activity record for the Opportunity. This workflow works fine when a new Opportunity is created directly using the browser interface, but it does not do so if I create an Opportunity record using the Apex method described above.

I would think the Process Builder shouldn’t care much about how the record is created, but something is certainly impacting this when I create the record in Apex. There may be some setting someplace that allows the Process Builder to work for any type of insert, but I'm not seeing it.

Any help on this is appreciated.