The Red Cross Ball came into existence before the Rose Ball.
The article below is a detailed look at balls in Monaco and the link to charities.The section on the Red Cross Ball, which is towards the end of the article, gives a flavour of how it earned its reputation as being, for many years, the social event of the summer on the French Riviera attracting the highest calibre of musical performers and A list guests . Something which is a bit hit and miss in recent years.
The history of Monaco Balls: magnificence and charity