Your taste buds will give the lie to this myth. The brown cremini have a deeper and more mushroomy flavor than the white mushrooms. Choose based on the needs of your recipe.
On a related note, did you know that portobello mushrooms are nothing more than large creminis?
They are the same species, Agaricus bisporus, but harvested at different stages of maturity. Brown are further along, eventually reaching the size and shape of what you know as portobello.
And yes, the flavor changes with maturity. They taste relatively similar, however, when you taste them next to other species of ‘shrooms.