What the difference of usage between the words statt and anstelle? Are they just synonyms?
The difference is your language style and feeling = they are synonyms. Together with "anstatt".
Example thx to comments:
Anstelle des Olivenöls nehme ich Rapsöl.
Statt des Olivenöls nehme ich einfach Rapsöl.
Anstatt des Olivenöls nehme ich einfach Rapsöl.
There is no difference in meaning in these three examples. In this short sentence I could even not say which I prefer and why. That comes usually with the characteristic style of the surrounding text.
There is a good and thorough explanation here (in German): https://www.duden.de/sprachwissen/sprachratgeber/statt-und-anstatt. "Statt/anstatt" can be used meaning "and not"; in this usage "anstelle" sounds odd.