The difference between them is as follows:
International Economics: Following the definition from the famous textbook by Krugman et al. (2017) the international economics is defined as a branch of economics that focuses on the special problems of economic interaction between sovereign states.
Financial Economics: According to W. Sharpe financial economics is a sub-field of economics that "concentrates on exchanges in which money of one type or another is likely to appear on both sides of a trade."
Why? The purpose of science is not just to come up with new ideas but to also organize them and categorize them. Thats the reason why we dont have just Physics but also astrophysics, geophysics, theoretical physics etc, why we dont have just biology but zoology, botany etc, why we dont have just psychology but clinical psychology, forensic psychology etc.
How? At some point the knowledge in any branch of science will become so wast that some scientists decide to focus only on special subset of the science and hence create sub-branch. For example, all physical sciences started as natural philosophy and all social sciences as moral philosophy but as science grew in complexity they started to diverge into different fields like physics, biology, economics or psychology, and as science grew ever more complex these fields further subdivided into their respective subfields.
Will a model of international economics suddenly become a model of financial economics if I simply change the names?
Its not about changing names. A model becomes an international economics model if it is applied to economic situation involving two or more sovereign states. A financial economics model will be a model thats applied to a situation where transactions involve money on both sides of a trade.
However, note the same models might be useful in different sub-fields. For example, laws of thermodynamics are widely used directly or indirectly in many branch of physics. A model that includes laws of thermodynamics but is a model pertaining to stellar phenomena will become an astrophysical model. A model based on laws of thermodynamics applied to some geothermal spot on earth will become geophysics model.
I saw myself repeating the same exact basic math and coming to the same conclusions, different names notwithstanding.
Yes, if a two subfields look at similar interactions its highly likely that they will use similar models and will result in similar conclusions. If anything I would be very worried if a scientific model/theory would work only in one narrow application and was unusable anywhere else. A hallmark of successful scientific theory is if it can explain a large number of phenomena with the least amount of effort. For example, marginal utility can explain a large number of phenomena whether you apply it labor economics, industrial organization, financial economics etc. Its a good sign when a scientific theory can be applied to multiple situations and yields similar conclusions (of course conditional on also testing those conclusions and making sure they are accurate).
Krugman, P. R., Melitz, M. J., & Obstfeld, M. (2017). International trade: theory and policy.