Short answerA brioche bun is rich, buttery and slightly sweet but can get greasy or fall apart under a juicy burger. A potato bun is soft, slightly sweet and sturdier, holding up better to sauce and juice. For balance and structure, potato usually wins; for indulgence, brioche.
Brioche
Enriched with butter and egg — rich, soft, a little sweet, golden. Lovely flavour, but it can turn greasy and, if it's too delicate, collapse under a wet, juicy burger.
Potato
Soft and pillowy from potato starch, slightly sweet, and notably sturdy — it resists sogginess and holds the burger together. The reliable workhorse.
Verdict
Want richness and don't mind a softer structure? Brioche. Want a soft bun that survives a juicy, saucy burger? Potato. Great burger joints often use potato for exactly that reliability.