When would you use a DA Form 4187 vs a DA Form 3838?

You use whichever you are told to use by the person/program you are applying to.

Every problem in the Army is solved by find the right banana to give to the right monkey to push the right button. First you find the right monkey, then you ask them which banana is the right one.

Every course, program, request etc in the Army has a person it is submitted to and that person has an example of what you need for that request. Go to them and ask them what you need to make this happen

The key point here for PERSONNEL ACTION within the facilities of the US Military such as Airborne School , etc. A DA Form 4187 is a particular form is known as a Personnel Action form. It is used by a soldier who wants to request a personnel action on their own behalf. This could include training, reassignment, extra rations, name changes, and more.
The DA Form 3838 is for an Application for Professional Training not at US Military facilities, such as Emergency Room Training at a Civilian Hospital or PhD School in line with an Officer's Duties such a Chaplains Conference or attending The Maneuver Center Conference at Fort Benning.