What is the difference between financial economics and finance?
Financial economics is what economics calls finance. Finance is what finance calls finance.
Less flippantly though, there's a long debate on whether finance is a subfield of economics, and this debate goes back at least to the PhD thesis of Markowitz. Prof. Milton Friedman famously opposed awarding Markowitz a PhD in economics from the University of Chicago for his portfolio theory because while Markowitz's work was brilliant, Friedman didn't consider it to be economics. I've heard Myron Scholes also express views that finance is to some extent distinct from economics.
Of course, you can find a huge range of views on whether finance is an economics subfield. Clearly much of the early work in finance was done by economists, and finance draws heavily from econometrics, time-series econometrics, and macroeconomics. On the other hand, risk-neutral pricing looks less like classic economics.
I'd personally argue there's some degree of economics claiming successful, related fields as its progeny. The mechanism design of Alvin Roth came out of operation research, but it's now been brought into economics.