Do stomach pain and stomach ache have the same meaning?
Yes, but stomach ache is far more commonly used in my experience.
Rover
For me there is a difference. A stomache ache is something that results from over-eating; a pain in my stomach is sharper, and may be caused by some illness or injury.
However, this may be just a personal thing; I note that OALD defines 'stomach ache' as 'pain in or near your stomach'.
As an ex-GP, I can confirm your usage. 'Pain' does have more of a connotation of severity, acuteness and probable pathology than does 'ache'.
Note: "Stomach" is often used by lay people for abdomen. Many (?most) abdominal aches/pains are not stomach-related. The generic medical term for these things is 'abdo pain'.
New (and newish) members may like to be informed (or reminded) that by entering this thread title (and many another one) into the search box at the very top of the page, you can find links to lots of other discussions on the subject.
There's an interesting one, for example, on the comparative usage of stomach ache/stomachache.
Rover
Yes. A 'general practitioner' or GP, at least in BrE/AusE, is normally understood to be a medical doctor.
However, you can be a general practitioner in law, architecture, or many other fields, and maybe that term is used within those professions without qualifying it. But lay people would generally assume you are a medical person if you say you're a GP.