Traditionally, financial engineers are people who create financial instruments, like contracts. Say you want to get a mortgage but instead of just paying interest you want to make it spicier somehow, something like “if the value of a home in this part of town is so and so, I’ll pay this, if not then that”, or something to that effect. A bank would come in with pricing, risk analysis, etc. for such a contract, done by people with knowledge in financial engineering. Nowadays the degree is chased mostly by people who want to get into investing, though, and they get an advantage in that due to the heavy quantitative focus of the program.
Econometrics is an area of statistics that is concerned with studying causality, particularly in economical and financial data. Say you think tariffs cause inflation (very topical lol), you would not only want to have theoretical basis for that, but an empirical foundation too, which is established by an econometric study.
I’m targetting financial engineering because it’s harder to get into, thus salaries are better, but would settle for econometrics because I can get similar opportunities if I play my cards right