Revisando, parecen sinónimos, pero no los puedes intercambiar así nada más.
En realidad, su uso depende del contexto.
Es algo como la diferencia que hay en español entre "concurso" y "competencia", que tampoco son intercambiables.

Contest: CHAMBERS -
A struggle for victory
Competition
Strife
debate

OXFORD
Contending
A competition

Competition
CHAMBERS
The act of competing: to seek or strive for something in opposition to others; to contend for a prize
Rivalry in strife for the same object
A match or trial of ability
OXFORD
event in which persons compete,: take part in contest, race, etc.
those competing whith one


To me, a contest is a type of competition, but not all competitions are contests

I would use “a competition” when talking about sports events, not “contest”
I would use “contest” for games like in a show on TV, or children games were you see which team wins

You really have to look at the context