What is the difference between vergeuden and verschwenden?
I believe they’re interchangeable. They are listed as synonyms on dict.cc, but of course there’s always the possibility that there’s some subtle difference that only a native speaker would appreciate.
They are synonyms.
My initial impulse was that “vergeuden” is about quality (not use wisely, spend for the wrong things, let go to waste, waste the potential), and “verschwenden” is about quantity (spend too much unnecessarily generously, so you lose it all), but my quick research did not produce real evidence for this. I guess in the end, whatever the difference, it is subtle and the effect is the same. Wiktionary and dwds list them as synonyms with very similar explanations.
Etymology-wise, duden says this about “vergeuden”:
mittelhochdeutsch vergiuden, zu: giuden = prahlen, großtun; prassen, wohl im Sinne von „den Mund aufreißen“
while wiktionary says this about “verschwenden”:
A causative formation derived from German verschwinden (“to disappear”)
Another interesting difference is the “Wortverlaufskurve” on dwds. Here you can clearly see that the popularity of “vergeuden” dropped while that of “verschwenden” went up. So “verschwenden” is the more modern word, and “vergeuden” is getting old fashioned. But both are still used, and especially together with “Zeit”, “vergeuden” is pretty common, although “Zeitverschwendung” is much more common than “Zeitvergeudung”, which I have never even seen used.