Trophies suit one winner. Medals suit everyone. If your event has a single champion and a presentation photo to take, a trophy earns its place. If your event has heats, age groups, divisions or a field of finishers, medals are the only award that scales without the budget falling over.
Most clubs end up doing both: one perpetual trophy that lives in the clubhouse and gets re-engraved each year, and a run of medals so every competitor takes something home.
A trophy is priced as an individual object, and the price climbs steeply with height and material. Medals are priced across a production run, so the more you order the less each one costs. That single fact decides most club budgets.
It also changes what you can promise entrants. Awarding every finisher is a genuine selling point for an event, and with medals it is affordable. With trophies it is not.
Trophies get kept when they are perpetual and live at the club. Individual trophies handed to juniors tend to end up on a shelf, then in a cupboard, then gone.
Medals get worn on the day, photographed, hung on a rack and kept for years. Rosettes are the cheapest option and the least kept, which makes them fine for very large participation events and poor for anything you want remembered.
Anything with volume: galas, opens, races, tournaments, sports days, gradings, presentation nights. Anything where every participant should walk away with something. Anything you want branded properly, because a bespoke medal carries your crest, your colours and your event name far better than an engraved plate on a plastic column.
And anything you will repeat. Once your design and tooling exist, next year's order is a quick reorder rather than a fresh decision.